DFT+U calculations of crystal lattice, electronic structure, and phase

THE JOURNAL OF CHEMICAL PHYSICS 135, 054503 (2011)

DFT+U calculations of crystal lattice, electronic structure, and phase stability under pressure of TiO2 polymorphs M. E. Arroyo-de Dompablo,1,a) A. Morales-García,2 and M. Taravillo2 1

Departamento de Química Inorgánica, MALTA Consolider Team, Facultad de CC. Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain 2 Departamento de Química Física I, MALTA Consolider Team, Facultad de CC. Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

(Received 21 January 2011; accepted 8 July 2011; published online 1 August 2011) This work investigates crystal lattice, electronic structure, relative stability, and high pressure behavior of TiO2 polymorphs (anatase, rutile, and columbite) using the density functional theory (DFT) improved by an on-site Coulomb self-interaction potential (DFT+U). For the latter the effect of the U parameter value (0 < U < 10 eV) is analyzed within the local density approximation (LDA+U) and the generalized gradient approximation (GGA+U). Results are compared to those of conventional DFT and Heyd-Scuseria-Ernzehorf screened hybrid functional (HSE06). For the investigation of the individual polymorphs (crystal and electronic structures), the GGA+U/LDA+U method and the HSE06 functional are in better agreement with experiments compared to the conventional GGA or LDA. Within the DFT+U the reproduction of the experimental band-gap of rutile/anatase is achieved with a U value of 10/8 eV, whereas a better description of the crystal and electronic structures is obtained for U < 5 eV. Conventional GGA/LDA and HSE06 fail to reproduce phase stability at ambient pressure, rendering the anatase form lower in energy than the rutile phase. The LDA+U excessively stabilizes the columbite form. The GGA+U method corrects these deficiencies; U values between 5 and 8 eV are required to get an energetic sequence consistent with experiments (Erutile < Eanatase < Ecolumbite ). The computed phase stability under pressure within the GGA+U is also consistent with experimental results. The best agreement between experimental and computed transition pressures is reached for U ≈ 5 eV. © 2011 American Institute of Physics. [doi:10.1063/1.3617244] I. INTRODUCTION

The overestimation of electron delocalization is a known drawback of density functional theory (DFT) methods, in particular, for systems with localized d-electrons and felectrons.1–3 The DFT+U method, developed in the 1990s,1, 4 combines the high efficiency of DFT with an explicit treatment of electronic correlation with a Hubbard-like model5 for a subset of states in the system. Non-integer or double occupations of these states is penalized by the introduction of two additional interaction terms, namely, the one-site Coulomb interaction term U and the exchange interaction term J. After the initial success with the rock-salt type MO family (M = Mn, Fe, Co, Ni),4, 6 the DFT+U has been extensively applied in the last years to investigate a wide variety of transition metal oxides where electron correlation results in a strong electron localization. It has been shown that DFT+U improves the capacities of DFT when dealing with energetic, electronic, and magnetic properties of insulating materials based on 3d transition metals (see, for instance, Refs. 2 and 7). Phase stability of that type of materials can also be successfully reproduced within the DFT+U, improving the results of the conventional DFT (see, for instance, the cases of MnO (Ref. 8) and FePO4 (Ref. 9). However, the adequacy of the DFT+U method to investigate early transition metal a) Author to whom correspondence should be addressed. Electronic mail:

[email protected]. Tel.: +34 91 3945222. FAX: +34 91 3944352.

0021-9606/2011/135(5)/054503/9/$30.00

compounds (Ti, V), where the more extended orbitals decrease the electron correlations, might be controversial.10–15 Insulating materials such as TiO2 provide an interesting case. TiO2 exhibits a large number of polymorphs as a function of pressure and temperature;16, 17 rutile, anatase, brookite, columbite, cotunnite, baddeleyite, and fluorite. This rich polymorphism, together with the interesting optical, electrical, and mechanical properties of TiO2 polymorphs, has originated a large number of ab initio investigations (see, for instance, Refs. 13, 14, 16, and 18–21). However, reproducing the phase stability of TiO2 forms is a remaining challenge for ab initio methods. Calculations performed at the Hartree-Fock (HF), density functional theory (generalized gradient approximation (GGA) and local density approximation (LDA) functionals), and hybrid functional (B3LYP, PBE0) levels have been shown to produce wrong relative stabilities for anatase and rutile polymorphs.17, 19 Correlation effects have been identified as a key factor to correctly reproduce the phase stability.17, 22 This context makes appealing to analyze the adequacy of the DFT+U method to study TiO2 polymorphs. The DFT+U method has been recently utilized to investigate electron transport in rutile-TiO2 ,15, 21 reduced forms of TiO2 ,13, 23 and ultrathin films of rutile-TiO2 .6, 12 In this work we focus on bulk phase stability and pressure driven transformations. Additionally, to complete previous investigations of TiO2 using the B3LYP and PBE0 hybrid functionals,19 we have also investigated the performance of the hybrid

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Arroyo, Morales, and Taravillo

Anatase [100]

J. Chem. Phys. 135, 054503 (2011)

Rutile [001]

Columbite [001]

FIG. 1. Schematic crystal structure of the TiO2 polymorphs under investigation; anatase, rutile, and columbite.

Heyd-Scuseria-Ernzerhof functional (HSE06) (Refs. 24–26) recently implemented in the Vienna ab initio simulation package (VASP). Figure 1 shows the crystal structure of the three polymorphs considered in this work, rutile, anatase, and columbite, which are all based on octahedral TiO6 units. In the rutile structure (hcp packing of oxygen atoms), each titanium octahedral shares two opposite edges along the c axis and vertices in the ab plane. In the anatase structure (ccp of oxygen atoms) the octahedra share four adjacent edges, forming zig-zag chains running along the a and b axis. The anatase and rutile varieties can be formed at ambient pressure. Early calorimetry works already determined the rutile form as the most stable one [H298 (anatase → rutile) = –11.7 kJ/mol (Ref. 27)], being the anatase a metastable form, at any temperature.28 The possibility of particle’s morphology affecting the thermodynamic stability of the TiO2 polymorphs has been investigated.28–30 It is nowadays recognized that anatase is the stable nanophase of TiO2 , that is, anatase gets thermodynamically stabilized by small particle size.28 Among the TiO2 polymorphs stabilized under high pressure conditions (columbite, baddeleyite, and cotunnite) we choose the columbite form (orthorhombic α-PbO2 ), in which oxygen atoms assume a distorted hexagonal close packing configuration. The TiO6 octahedra form edge-sharing chains parallel to the c axis. Individual chains are connected by corner-sharing to form a three-dimensional framework. High pressure-high temperature treatment of either anatase or rutile TiO2 yields the columbite modification, which can be quenched to ambient conditions.31–33 One problem encountered using the DFT+U method is the determination of an appropriate U parameter value for each compound. The values of U can be determined through a linear response method which is fully consistent with the definition of the DFT+U Hamiltonian, making this approach for the potential calculations fully ab initio.34 An alternative route consists of selecting these values so as to account for the experimental results of physical properties: magnetic moments,35 band gaps,11 redox potentials,2, 11 or reaction enthalpies.36 Previous DFT + U works found a value of U = 10 eV for rutile TiO2 by fitting calculated band gaps to the experimental value.15, 21 In this work we discuss the suitability of the DFT+U method utilizing the GGA+U and the LDA+U functionals to investigate TiO2 polymorphs and how the U parameter (3 eV < U < 10 eV) affects various properties of TiO2 ; crystal lattice (Sec. III A),

electronic structure (Sec. III B), relative stability (Sec. III C) and pressure driven phase transformations (Sec. III D). Results are compared to the performance of the conventional DFT and the hybrid HSE06 functionals. We will show that the DFT+U method and the hybrid HSE06 functional are suitable to investigate crystal structure and electrical properties of TiO2 polymorphs, improving the accuracy of conventional DFT for band gaps and lattice parameters prediction. Further, while the conventional DFT, the hybrid functionals, and the LDA+U fail to reproduce the phase stability at both ambient and high pressure, proper results are obtained within the GGA+U method. To the best of our knowledge, it is the first time that the suitability of the DFT+U to investigate TiO2 -polymorphism is discussed and demonstrated. II. METHODOLOGY

The total energy calculations and structure relaxations were performed with the VASP.37, 38 First, calculations were done within the DFT framework with the exchangecorrelation energy approximated in the GGA and in the LDA, utilizing two sets of potentials; ultrasoft-pseudopotential (PP) and projector augmented wave (PAW).39 For the latter we tested two different GGA-functionals, PBE (Ref. 40) and PW91,41 which yielded very similar results. Our results utilizing the PP are consistent with those previously reported; in short, the method fails to reproduce the relative energy of polymorphs. Thus, for conciseness, in this work we will only discuss the PAW results.39 The LDA and the PBE form of the GGA exchangecorrelation functionals have been used together with their LDA+U and GGA+U variants as implemented in VASP. The Ti(3p, 3d, 4s) and O (2s, 2p) were treated as valence states. Test calculations performed including the Ti-3s as valence states yielded equivalent results. DFT+U calculations were performed following the simplified rotationally invariant form proposed by Dudarev.38, 42 Within this approach, the onsite Coulomb term U and the exchange term J, can be grouped together into a single effective parameter (U-J) and this effective parameter will be simply referred to as U in this paper. The J value was fix to 1 eV. Effective U values of U = 3, 5, 8, and 10 eV were used for the Ti-3d states. The energy cut off for the plane wave basis set was kept fix at a constant value of 600 eV throughout the calculations. The reciprocal space sampling was done with k-point Monckhorst-Pack


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DFT+U calculations of TiO2 polymorphs

J. Chem. Phys. 135, 054503 (2011)

grids of 6 × 6 × 8 for rutile, 8 × 8 × 8 for anatase, and 10 × 10 × 8 for columbite. As a first step the structures were fully relaxed (cell parameters, volume, and atomic positions) and the final energies of the optimized geometries were recalculated so as to correct the changes in the basis set of the wave functions during relaxation. Second, from the relaxed structure, within the GGA+U approximation, calculations were performed at various constant volumes and the energyvolume data were fitted to the Birch-Murnaghan equation of state:43 9V0 B0 2/3 B0 [x − 1]3 E(V ) = E0 + 16 +[x 2/3 − 1]2 [6 − 4x 2/3 ] , (1) where x = (V0 /V), being V and V0 the volume at pressure p and the equilibrium volume at ambient pressure, respectively. B0 and B0 are the bulk modulus at ambient pressure and its pressure derivative, respectively, and E0 is the equilibrium energy.

Hybrid calculations were performed using the HSE06 functional, as implemented in VASP, with a screening parameter μ = 0.2 Å−1 . The NKRED parameter was set to 2 meaning that a HF kernel was evaluated on a coarser k-point grid.26 Full structure optimization was performed (cell parameters, volume, and atomic positions). The optimized structures of anatase and rutile were used as input for accurate static calculations (the NKRED parameter was omitted, and a normal grid was utilized in the HF calculation). III. RESULTS AND DISCUSSION A. Crystal structure

Table I compares the calculated lattice parameters and volume for the fully relaxed structures of TiO2 polymorphs (HSE06, GGA/LDA , and GGA+U / LDA+U with U = 3, 5, 8, and 10 eV) with the experimental ones. The unit cell volume is overestimated in about 3% within the GGA. The overestimation enlarges within the GGA+U, ranging from

TABLE I. Unit cell parameters (in Å) and volume per formula unit (in Å3 ) for TiO2 polymorphs obtained after complete structural optimization with HSE06, conventional GGA/LDA, and GGA+U /LDA+U compared to experiments. Numbers in parentheses indicate the percent of deviation from experimental data. Polymorph ANATASE

RUTILE

COLUMBITE

Method HSE06 GGA GGA+U GGA+U GGA+U GGA+U LDA LDA+U LDA+U LDA+U LDA+U Experiment31 HSE06 GGA GGA+U GGA+U GGA+U GGA+U LDA LDA+U LDA+U LDA+U LDA+U Experiment (Refs. 31 and 63) HSE06 GGA GGA+U GGA+U GGA+U GGA+U LDA LDA+U LDA+U LDA+U LDA+U Experiment (Ref. 31.)

U (eV)

3 5 8 10 3 5 8 10

3 5 8 10 3 5 8 10

3 5 8 10 3 5 8 10

V (Å3)

a (Å)

34.07 (0) 35.13 (3.11) 36.15 (6.10) 36.81 (8.04) 37.81 (10.9) 38.46 (12.9) 33.29 (–2.28) 34.24 (0.50) 34.84 (2.26) 35.82 (5.14) 36.50 (7.13) 34.07

b (Å)

c (Å)

(c/a)

3.766 (–0.50) 3.807 (0.58) 3.853 (1.79) 3.881 (2.53) 3.922 (3.62) 3.947 (4.28) 3.74 (–1.19) 3.793 (0.21) 3.819 (0.89) 3.86 (1.98) 3.882 (2.56) 3.78512(8)

9.609 (1.02) 9.693 (1.90) 9.742 (2.42) 9.774 (2.76) 9.830 (3.34) 9.871 (3.77) 9.521 (0.09) 9.52 (0.08) 9.555 (0.45) 9.617 (1.10) 9.687 (1.84) 9.51185(13)

2.551 (1.51) 2.545 (1.27) 2.529 (0.64) 2.518 (0.20) 2.506 (–0.28) 2.501 (–0.48) 2.546 (1.31) 2.51 (–0.12) 2.502 (–0.44) 2.491 (–0.87) 2.495 (–0.72) 2.513

31.08 (–0.45) 32.10 (2.82) 32.87 (5.28) 33.41 (7.01) 34.17 (9.45) 34.69 (11.1) 30.32 (–2.88) 31.12 (–0.32) 31.68 (1.47) 32.33 (3.55) 32.92 (5.44) 31.22

4.590 (–0.08) 4.650 (1.22) 4.671 (1.68) 4.687 (2.03) 4.709 (2.51) 4.725 (2.86) 4.556 (–0.82) 4.580 (–0.30) 4.599 (0.11) 4.619 (0.55) 4.637 (0.94) 4.5938(1)

2.950 (–0.29) 2.968 (0.32) 3.012 (1.80) 3.042 (2.82) 3.081 (4.13) 3.108 (5.05) 2.922 (–1.24) 2.967 (0.28) 2.995 (1.23) 3.031 (2.45) 3.062 (3.49) 2.9586(1)

0.642 (–0.31) 0.638 (–0.93) 0.645 (0.16) 0.649 (0.78) 0.654 (1.55) 0.658 (2.17) 0.641 (–0.46) 0.648 (0.62) 0.651 (1.09) 0.656 (1.86) 0.66 (2.48) 0.644

30.52 (–0.23) 31.51 (3.00) 32.23 (5.36) 32.77 (7.13) 33.56 (9.71) 34.09 (11.4) 29.78 (–2.65) 30.53 (–0.20) 31.05 (1.50) 31.83 (4.05) 32.35 (5.75) 30.59

4.539 (–0.04) 4.579 (0.84) 4.607 (1.45) 4.629 (1.94) 4.658 (2.58) 4.674 (2.93) 4.506 (–0.77) 4.526 (–0.33) 4.542 (0.02) 4.572 (0.68) 4.591 (1.10) 4.541(6)

4.893 (–0.26) 4.936 (0.61) 5.008 (2.08) 5.053 (3.00) 5.115 (4.26) 5.151 (4.99) 4.850 (–1.14) 4.925 (0.38) 4.971 (1.32) 5.030 (2.53) 5.066 (3.26) 4.906(9)

1.077 (–0.28) 1.078 (–0.18) 1.087 (0.65) 1.092 (1.11) 1.098 (1.67) 1.102 (2.04) 1.076 (–0.37) 1.088 (0.74) 1.094 (1.30) 1.1 (1.85) 1.103 (2.13) 1.080

5.495 (0.04) 5.576 (1.51) 5.585 (1.67) 5.603 (2.00) 5.635 (2.59) 5.664 (3.11) 5.450 (–0.78) 5.477 (–0.33) 5.501 (0.02) 5.537 (0.68) 5.563 (1.10) 5.493(8)


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J. Chem. Phys. 135, 054503 (2011)

∼5% for U = 3 eV to 12.5% for U = 10 eV. Deviation of the lattice parameters with respect to the experimental values is of the order of 1.5% within the conventional GGA, and increases within the GGA+U as a function of U, reaching a maximum difference at U = 10 eV of around 4%. Within the conventional LDA the unit cell volume and lattice parameters are underestimated. The effect of the Hubbard U is to expand the unit cell, and a good reproduction of the experimental crystal cell is achieved at U values below 5 eV. The HSE06 underestimates the unit cell volume by a small amount of the order of 0.3%. As it is generally found,25, 26, 44 the HSE06 predicts more accurate cell parameters, with average deviations below 0.4%. The good performance of the hybrid functionals to reproduce the crystal structure of TiO2 polymorphs was pointed out by Labat and co-workers, who found a deviation of lattice parameters of 0.3% for rutile and 1.1% for anatase utilizing the PBE0 hybrid functional.19 In order to evaluate the effect of the U parameter in the crystal structure prediction, it is more effective to analyze lattice parameters ratios. The c/a ratio is a crucial parameter for the rutile structural type, since changes in the c/a ratio depend on the formation of metal-metal bonds45, 46 (see discussion in Sec. III B). Figure 2 plots the variation of the c/a ratio for the different polymorphs versus the Hubbard-U. The experimental values are indicated by a horizontal line. HSE06 functional -1 0 1 2 3 4 5 6 7 8 9 10 11 2.55 2.54 2.53

Anatase

2.52

Experimental 2.51 2.50 2.49

Calculated c/a

0.660 0.655

GG A HSE0 6 LD A

0.650

Rutile

0.645

Experimental

values are denoted by stars. For rutile and columbite, where the TiO6 octahedra share edges along the c axis, the c/a ratio increases with the value of U. The opposite trend is observed for the anatase polymorph, in which edge-sharing occurs in the ab plane (see Fig. 1). It can be observed in Fig. 2 that independent of the functional (LDA or GGA), introducing a U correction term allows a better description of the c/a ratio for the three polymorphs. Within the GGA+U for columbite and anatase small U values of 0.7 and 2.6 eV, respectively, match the experimental value. The required U values to reproduce the experimental data are similar in the LDA+U (0.82 eV columbite, 1.2 eV rutile). Yet, for these two polymorphs the HSE06 provides a quite accurate value. For the anatase polymorph, HSE06 and GGA/LDA show larger deviations to the experimental c/a ratio, and U values of 6 eV (GGA+U) and 3 eV (LDA+U) are required to match experimental data. The worse performance of DFT/HSE06 for the anatase polymorph falls in the context of the difficulties found by any first principle methods to reproduce this structure, compared to the rutile for which lower deviations to experimental values have been observed.17, 19 Worth mentioning, while adequate U values improve the results obtained within the conventional DFT, very large U values lead to a poor crystal lattice prediction. Note the large deviation for the columbite polymorph when U > 6 eV. B. Electronic structure of anatase and rutile

One of the most common approaches to determine the appropriate value of U is to compare the calculated band gaps for a set of U values with the experimental band gap. This practice, however, should be exercised with caution, unless computational methods designed for describing excited states are utilized. Therefore, in this work rather than attempting to extract a concrete U value, we discuss the general features observed in the calculated electronic structure when correlation effects are considered. Experimental band gaps are 3.03 and 3.2 eV for rutile47 and anatase,48 respectively. Figure 3 shows the band gap values extracted in this work 0

0.640 0.635 1.105 1.100 1.095 1.090

Columbit e

1.085

Ex perimental

1.080 1.075

-1 0 1 2 3 4 5 6 7 8 9 10 11

U (eV) FIG. 2. Calculated DFT+U values for the c/a ratio for the optimized crystal structures of TiO2 polymorphs as a function of the U parameter (for conventional DFT U = 0 eV); LDA+U (circles) and GGA+U (squares). Stars correspond to the HSE06 values. Lines are a guide to eye. Experimental data are indicated by horizontal lines.

Calculated Band Gap (eV)

3.6

2

4

6

8

10

Anatase

3.4 3.2 3.0 2.8 2.6

Rutile

Anatase Rutile

2.4 2.2 2.0 1.8

Anatase GGA Rutile GGA Experimental HSE06 Anatase LDA Rutile LDA

0

2

4 6 U (eV)

8

10

FIG. 3. Calculated DFT+U band-gap for anatase (squares) and rutile (triangles) TiO2 polymorphs as a function of the U parameter; LDA+U hollow symbols and GGA+U filled symbols. Lines are a guide to eye. Stars correspond to the HSE06 values. Experimental data are indicated by horizontal red bars.


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8

Energy (eV)

6 4

e

J. Chem. Phys. 135, 054503 (2011)

TOTAL Ti (d) O (p)

M- O σ*

g

t

2

2g

M- O π* M- M

Ti (eg) dxy dxz

II c

Ti (t2g)

dyz

0 -2

M- O π

-4

M- O σ

-6

O (p)

Calculated DOS

FIG. 4. Calculated density of states (DOS) of rutile-TiO2 within the conventional GGA method. The black lines denote the total DOS; the partial DOS of titanium and oxygen are represented in green and red lines, respectively. The Fermi level has been arbitrarily chosen as the origin of the energy. The right panel is an schematic representation of rutile-TiO2 band structure adapted from Refs. 46 and 50.

from the calculated density of state (DOS) for rutile (squares) and anatase (triangles). Calculated band gaps within the conventional DFT (rutile ∼ 1.86 eV, anatase ∼ 2.07 eV) are similar to those obtained in previous DFT studies.19, 20 Hybrid functionals yielded too large band-gaps, for instance, in Ref. 19 the PBE0 values are 4.02 eV and 4.50 eV for anatase and rutile, respectively. Interestingly, Janotti et al. reported a HSE06 band gap of 3.05 eV after they reduced the fraction of exact exchange in the hybrid functional to 20% from the original 25%.49 Using the HSE06 functionals we obtained band gap values of 3.6 eV for anatase and 3.2 eV for rutile (see stars in Fig. 3). Within the DFT+U the band gap increases with the value of the U parameter, confirming the fact that a Hubbard-like correction term (U) substantially improves the accuracy of the calculated band-gap, compared to the conventional DFT. In good agreement with other authors,15, 21 values of 10 and 8.5 eV are required to match the experimental band gap of rutile and anatase, respectively. However, these values seem too large if one takes into account U values extracted for other transition metal oxides where electron localization effects are more relevant (for NiO U = 6.4 eV (Ref. 6), for MnO U = 4.4 eV (Ref. 6)). Therefore, other features in the electronic structure than the band gap should be examined. Figure 4 shows the total calculated DOS of rutile together with the partial DOS of Ti and O in green and red, respectively. The Fermi level is arbitrarily chosen as the origin of the energy. The DOS can be interpreted according to the schematic band structure depicted on the right panel (adapted from Refs. 46 and 50). An isolated Ti4+ ion contains no occupied d orbitals, and therefore for a purely ionic Ti–O interaction no Ti-d states should be occupied. This means that any d character in the DOS below the Fermi level is a direct result of oxygen–titanium covalent interactions. The σ overlap between Ti-3d(eg ) and O-2p orbitals results in the bonding σ band, which appears below the Fermi level (predominantly oxygen-2p in character), and the σ * band above the Fermi level (mostly consisting of titanium-3d states). The significant mixing of O-2p states and Ti-3d states in both the σ and σ *

bands supplies direct evidence for the strong covalent interaction of the Ti–O bonds. The titanium t2g orbitals (dxy , dxz , and dyz ) are involved in π interactions with the filled oxygen p orbitals of appropriate symmetry. The bonding π -states appear in the valence band and the antibonding π * in the conduction band. In a perfectly cubic octahedral environment the t2g orbitals are degenerate. In rutile, however, the TiO6 octahedra share edges along the c axis, and vertex in the ab plane. This distortion breaks the degeneration of the t2g orbitals. Given the short Ti–Ti distance along the [001] axis (d(Ti–Ti) = c = 2.958 Å), the dxy orbitals interact forming a metal σ -band along the shared edges of the octahedra. Figure 5 shows the evolution of the DOS with the U parameter within the GGA+U. A similar evolution is observed within the LDA+U (see Ref. 51). Increasing the U value has two effects: (i) opening the energy gap between the valence and conduction bands and (ii) a progressive merging of the t2g and eg derived bands, which do not differentiate any longer at U = 10 eV. The U parameter keeps the d orbitals atomic like, diminishing the effective overlapping of O-2p and Ti-3d orbitals. In rutile, the downshift of the σ * band is related to a reduction in the Ti(eg )-Op overlapping (weaker Ti–O bond, longer a lattice parameter), while the up shift and narrowing of the t2g -derived band is associated to a weaker Ti–Ti interaction (longer c lattice parameter). Changes of the t2g band as a function of U are more pronounced that those of the eg band as the c lattice parameter increases more than the a parameter (c/a ratio increases). Therefore, the DOS modifications as a function of the U parameter are consistent with the evolution of the lattice parameters and the c/a ratio (see Table I and Fig. 2). To further analyze the adequacy of the DFT+U method, in a first approximation, the calculated DOS could be qualitatively compared with experimental x-ray absortion spectra and electron energy loss spectra. It is, however, important to point out that such comparison neglects the excitation aspects, which can be taken into account by means of quasiparticle calculations in the GW-approach (see, for instance, Ref. 51 and references therein). The measured



-4

-2

U = 10 eV

2

4

6

8

CFS = 0.9 eV

CFS = 1.2 eV

U = 3 eV

CFS = 1.9 eV

0.12 0.09 0.06 0.03 0.00 -0.03 -0.06 -0.09

0

GG A

-4

2

4

6

8

10

U (eV) CFS = 2.4 eV

HSE06 -6

E rut < E anat < E col

JANAF 1973

CFS = 1.6 eV

GGA+U

Anatase GGA Columbite GGA Anatase LDA Columbite LDA Anatase HSE06 Columbite HSE06 Exp. Anatase

0.15

Rao 1961

U = 5 eV

0.18

Ranade 2002 Levchenko 2006

Ca lc ulated DOS of RutileTi O2

U = 8 eV

0

Less stable than Rutile

-6

J. Chem. Phys. 135, 054503 (2011)

Calculated energy difference (eV/f.u.)

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CFS = 2.6 eV

-2

0

2

4

6

8

Energy (eV) FIG. 5. DOS of rutile TiO2 calculated using the HSE06, the conventional GGA method, and the GGA+U with U values of 3, 5, 8, and 10 eV. Guidelines show the displacement of the bands as the value of U increases. CFS refers to the crystal field splitting between averaged energies of t2g and eg derived bands. The DOS calculated using the LDA+U is shown in the supplementary information.

energy difference within the average energies of the t2g and eg bands (crystal field splitting, CFS) varies from 1.8 to 3.0 eV.53, 54 It can be seen in Fig. 5 that a too large U value (U > 5 eV) dramatically reduces the CFS (for LDA+U results see Ref. 51). Similarly, for anatase-TiO2 (experimental CFS value 2.6 eV (Ref. 54)), the introduction of the U parameter in the calculation reduces the CFS value; 2.2 eV for GGA, 1.7 eV for GGA+U, U = 5 eV (see Ref. 51). In short, the DFT+U method allows a good qualitative description of the electronic structure for moderate U values, below 5 eV. Using these results as a starting point, the U value could still be optimized to give the best agreement with available experimental data utilizing the GW method based on the DFT+U.52 C. Phase stability

Rutile is the stable bulk phase of TiO2 at ambient pressure,29 with the enthalpy of the anatase to rutile transition taking values ranging from +0.42 kJ/mol (Ref. 55) to –11.7 kJ/mol.27 However, previous DFT works found anatase as the most stable phase.17–19 This discrepancy could not be solved by testing different exchange-correlation functionals.17, 19 Hybrid functionals also produce wrong phase stability, regardless the amount of HF exchange included.19 Figure 6 represents the calculated total energy differences, referred to the rutile polymorph, for anatase and columbite within the DFT and DFT+U (U = 3, 5, 8, and 10 eV), and using the HSE06

FIG. 6. Calculated DFT+U total energy differences for TiO2 polymorphs as a function of the U parameter; LDA+U hollow symbols and GGA+U filled symbols. Data are referred to the rutile polymorph. Eanatase –Erutile is denoted by squares (DFT+U) and star (HSE06). Erutile –Ecolumbite is denoted by triangles (DFT+U) and pentagon (HSE06). Solid lines are a guide to eye. The red horizontal bars correspond to experimental data for –H of the anatase to rutile transformation taken from the references. Also see Refs. 29, 30, 61, and 62. The light grey area indicates the U values for what the GGA+U reproduces the experimental phase stability.

functional. Positive energy differences indicate that a polymorph is less stable than the rutile form. Experimental data of H (rutile → anatase) are indicated by horizontal red lines. The conventional GGA gives a relative energy Eanatase < Erutile , which is in clear disagreement with experiments.28 The LDA produces a complete wrong stability sequence, Ecolumbite < Eanatase < Erutile , with the columbite form as the most stable polymorph. The HSE06 functional predicts that the anatase form is 0.086 eV/f.u. more stable than the rutile form; this energy difference is reduced to 0.055 eV/f.u in more accurate calculations (see Sec. II). Next, we analyze the effect of the U parameter on phase stability. It can be seen in Fig. 6 that the relative energy of the polymorphs strongly depends on the value of U. Noteworthy, the LDA and GGA functionals predict different energetic stability. Within LDA+U the stability sequence is dominated by the strong stabilization of the columbite form, a phase only reachable at high pressure of the order of 6 GPa. Even though at U = 2.5 eV the rutile becomes the most stable form, the columbite remains more stable than the anatase form. This is to say, for U > 2.5 eV the energetic sequence is Erutile < Ecolumbite < Eanatase , which is not consistent with the observed stability for the bulk phases. It can be seen in Fig. 6 that the experimental results (Erutile < Eanatase < Ecolumbite ) are well reproduced within the GGA+U in the range of 5 eV < U < 8 eV. For values of U > 8 eV, the rutile is the most stable polymorph but the columbite form gains in stability to the anatase form (E rutile < Ecolumbite < Eanatase ). To investigate phase stability under pressure, several fixed-volume calculations were performed starting from the GGA and GGA+U optimized structures. The strong failure of the LDA and hybrid functionals to reproduce phase stability at ambient pressure, at the expense of a large computational effort for the latter, discourages the investigation un-


054503-7


J. Chem. Phys. 135, 054503 (2011)

ANATASE

RUTILE

-25.8

DFT

-26.0

Calculated Energy (eV/f.u.)

COLUMBITE

-23.6

U = 3 eV

-23.8

-26.2

-24.0

-26.4

-24.2

-26.6

-24.4

-26.8

-24.6

-27.0

-24.8 26

28

30

32

34

36

38

40

42

26

28

30

32

34

36

38

40

42

-22.4 -19.4

U = 5 eV

-22.6

U = 10 eV -19.6

-22.8 -19.8 -23.0

-20.0

-23.2

-20.2

-23.4

-20.4 26

28

30

32

34

36

38

40

42

26

28

30

32

34

36

38

40

42

3

Volume ( Å /f.u.) FIG. 7. Total energy vs. volume curves for TiO2 polymorphs. Symbols correspond to the conventional GGA and GGA+U calculated data, and lines show the fitting to the Birch-Murnaghan equation of state.

der pressure. Figure 7 shows the calculated total energy as a function of the volume for the different polymorphs within the GGA and GGA+U (U = 3, 5, and 10 eV), together with the corresponding fit to the Birch-Murnaghan equation of state. Table II lists the parameters of the fits of the ab initio energyvolume data for the investigated TiO2 forms. The bulk moduli are clearly underestimated, independent of the choice of U. Note the major differences with experimental bulk moduli are observed for U = 10 eV. For anatase and rutile in-

creasing U values lead to lower bulk moduli, a trend observed in other systems: Lu2 O3 ,56 FeS2 ,57 and MnO.8 The fluctuations of the bulk moduli as a function of U obtained for the Columbite phase are not surprising neither (see, for instance, Ce2 O3 (Ref. 58) or Fe1–x Mgx O (Ref. 59)). In Fig. 7 it can be observed that, within the GGA, the global energy minimum, this is to say the polymorph stable at ambient pressure, corresponds to anatase, with rutile being a metastable phase at any pressure. The GGA+U corrects

TABLE II. Calculated equation of state parameters for TiO2 polymorphs (conventional GGA vs. GGA+U). E0 , V0 , B0 , and B0 are the zero-pressure energy, volume, bulk modulus, and its pressure derivative, respectively. Method

U (eV)

E0 (eV/f.u.)

V0 (Å3 /f.u.)

B0 (GPa)

B0

rms (eV/f.u.)

GGA GGA+U GGA+U GG+U Experiment (Ref. 31.)

3 5 10

–26.9045 –24.7312 –23.3733 –20.2905

35.272 36.239 36.902 38.572 34.07

169.9 164.9 162.3 157.6 179 ± 2

2.27 2.53 2.61 2.67 4.5

0.2 0.2 0.3 0.4

GGA GGA+U GGA+U GGA+U Experiment (Ref. 63.)

3 5 10

–26.8128 –24.6944 –23.3727 –20.3735

32.186 32.946 33.458 34.740 31.22

200.4 199.5 198.7 195.3 211 ± 7

4.98 4.77 4.62 4.40 6.76

0.7 0.6 0.6 0.5

GGA GGA+U GGA+U GGA+U Experiment (Ref. 31.)

3 5 10

–26.8288 –24.6718 –23.3363 –20.3171

31.651 32.334 32.866 34.179 30.59

195.9 206.8 202.3 193.9 258 ± 8

4.22 3.62 3.69 3.75 4.1

1.1 0.3 0.4 0.3

Polymorph ANATASE

RUTILE

COLUMBITE

rms =

(E−Ef it )2 n

.


054503-8


0.12

Rutile Anatase Columbite

0.08

Calculated Enthalpy difference (eV/f.u.)

J. Chem. Phys. 135, 054503 (2011)

0.04

U = 10 eV

rut-col

Ptr

= 16 GPa

0.00 -0.04 0

2

4

6

8

10

12

0.06

16 18

U = 5 eV

0.04

ana-col

Ptr

0.02

= 1.5 GPa rut-col

Ptr

0.00 -0.02

14

ana-rut

Ptr

-0.04 0

= 10.1 GPa

= 0.03 GPa

2

4

6

8

10

12

0.06 0.04 0.02

14

16 18

U = 3 eV ana-col

Ptr

= 2.5 GPa rut-col

Ptr

0.00 ana-rut

-0.02

Ptr

-0.04 0

2

= 6.1 GPa

= 1.8 GPa

4

6

8

10

12

14

16 18

Pressure (GPa) FIG. 8. Calculated Enthalpy vs. pressure for TiO2 polymorphs within the GGA+U (U = 3, 5, and 10 eV).

these discrepancies with experiments. For U = 3 eV the rutile form is stabilized by pressure, and for U = 5, 10 eV the rutile becomes the stable polymorph at ambient pressure, in good agreement with experimental results. The curve of the columbite polymorph crosses that of the most stable phase (anatase for GGA and U = 3 eV, rutile for U > 5 eV) at a certain volume, indicating that columbite becomes more stable at a sufficient pressure. Comparing the predicted pressure of the anatase → columbite and rutile → columbite transformations with experimental data is another way to determine appropriate U values.

D. Transition pressures

Figure 8 shows the calculated enthalpy-pressure variation for the TiO2 polymorphs (at 0 K), within the GGA+U. The effect of U is to extend the stability field of the rutile phase, raising/decreasing the pressure of the rutile/anatase to columbite transformations. Experimentally, it has been observed that anatase transforms to columbite at 2.6–7 GPa,31, 32 being the pressure of the transformation highly dependent on the particle size. Rutile undergoes the transformation to columbite at about 10 GPa.33, 60 Therefore, the GGA+U with U = 5 eV provides the best agreement with experiments (PA-C = 1.5 GPa, PR-C = 10 GPa). At U = 10 eV the rutile transformation occurs at too high pressure (16 GPa), and the anatase to columbite transformation is not feasible. IV. CONCLUSIONS

The suitability of DFT+U methodology to investigate TiO2 polymorphs is discussed and compared to conventional DFT and hybrid HSE06 functionals. The GGA+U

and LDA+U methods improve the prediction of ground state properties and electronic structure of the investigated TiO2 polymorphs (anatase, rutile and columbite), with respect to the conventional GGA/LDA. As expected, it is not possible to extract an universal value of the U parameter to reproduce all TiO2 properties. Reproducing the experimental band gap of rutile-TiO2 requires a U parameter of 10 eV, which in turn produces a band structure in disagreement with experiments. In addition, such large U values worsen the reproduction of crystal structure. Generally speaking, we found that the DFT+U (U ≈ 5 eV) method is well suited to investigate the properties of individual TiO2 polymorphs (anatase, rutile, and columbite). Yet, the hybrid HSE06 performs equally well to accurately reproduce the crystal and electronic structures of TiO2 polymorphs. We found that conventional GGA/LDA and HSE06 fail to reproduce TiO2 -phase stability, yielding a too low energy for the anatase polymorph (Eanatase < Erutile ). Introducing a Hubbard correction term (U) in the conventional LDA (LDA+U) has the effect of excessively stabilizing the columbite polymorph, which is predicted as more stable than the anatase form at any U value. Note that contrary to any experimental observation columbite is even more stable than rutile for U < 2.6 e V. The GGA+U method yields an energetic sequence consistent with experiments, Erutile < Eanatase < Ecolumbite , for U values between 5 and 8 eV. The GGA+U method also allows a correct prediction of phase stability at high pressure. Phase transformations under pressure are the best reproduced for U values of the order of 5 eV. For U = 5 eV predicted transition pressures are rutile → columbite 10 GPa (expt. 10 GPa) and anatase → columbite 1.5 GPa (expt. 2.5–7 GPa). The present results indicate that the treatment of correlation effects is important to investigate the phasestability of TiO2 . We conclude that computational results for titania polymorphs are very sensitive to the choice of the functional (LDA/GGA) and the treatment of correlation effects (U value); extreme care should be exercised when selecting a computational methodology to investigate this system, in particular, for phase stability.

ACKNOWLEDGMENTS

Financial resources for this research were provided by the Spanish Ministry of Science (MAT2007–62929, CSD2007– 00045, CTQ2009–14596-C02–01) and Project No. S2009PPQ/1551 funded by Comunidad de Madrid. A.M.-G. acknowledges a grant from the FPI Program of Spanish Ministry of Science. Valuable comments from D. Morgan, J. Tortajada, and V. G. Baonza are greatly appreciated. M.E.AdD. thanks R. Armiento and G. Ceder for their kind help with the HSE06 calculations. 1 V.

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