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THE JOURNAL OF CHEMICAL PHYSICS 135, 054503 (2011)
DFT+
U
calculations of crystal lattice, electronic structure, and phase
stability under pressure of TiO2polymorphs
M. E. Arroyo-de Dompablo,1,a) A. Morales-García,2and M. Taravillo2
1Departamento de Química Inorgánica, MALTA Consolider Team, Facultad de CC. Químicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain
2Departamento 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 behav-
ior of TiO2polymorphs (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
Uparameter 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 conven-
tional DFT and Heyd-Scuseria-Ernzehorf screened hybrid functional (HSE06). For the investigation
of the individual polymorphs (crystal and electronic structures), the GGA+U/LDA+Umethod and
the HSE06 functional are in better agreement with experiments compared to the conventional GGA or
LDA. Within the DFT+Uthe reproduction of the experimental band-gap of rutile/anatase is achieved
with a Uvalue of 10/8 eV, whereas a better description of the crystal and electronic structures is ob-
tained 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+Uexcessively
stabilizes the columbite form. The GGA+Umethod corrects these deficiencies; Uvalues 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+Uis also consistent with
experimental results. The best agreement between experimental and computed transition pressures is
reached for U≈5eV.© 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 f-
electrons.1–3The DFT+Umethod, developed in the 1990s,1,4
combines the high efficiency of DFT with an explicit treat-
ment of electronic correlation with a Hubbard-like model5for
a subset of states in the system. Non-integer or double occu-
pations of these states is penalized by the introduction of two
additional interaction terms, namely, the one-site Coulomb
interaction term Uand the exchange interaction term J. Af-
ter the initial success with the rock-salt type MO family (M
=Mn, Fe, Co, Ni),4,6the DFT+Uhas been extensively ap-
plied in the last years to investigate a wide variety of tran-
sition 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 3dtransition metals (see, for instance, Refs. 2and
7). Phase stability of that type of materials can also be suc-
cessfully reproduced within the DFT+U, improving the re-
sults of the conventional DFT (see, for instance, the cases of
MnO (Ref. 8) and FePO4(Ref. 9). However, the adequacy
of the DFT+Umethod to investigate early transition metal
a)Author to whom correspondence should be addressed. Electronic mail:
e.arroyo@quim.ucm.es. Tel.: +34 91 3945222. FAX: +34 91 3944352.
compounds (Ti, V), where the more extended orbitals de-
crease the electron correlations, might be controversial.10–15
Insulating materials such as TiO2provide an interesting
case.
TiO2exhibits a large number of polymorphs as a func-
tion of pressure and temperature;16,17 rutile, anatase, brookite,
columbite, cotunnite, baddeleyite, and fluorite. This rich poly-
morphism, together with the interesting optical, electrical, and
mechanical properties of TiO2polymorphs, 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 TiO2forms is a remaining challenge for ab initio
methods. Calculations performed at the Hartree-Fock (HF),
density functional theory (generalized gradient approxima-
tion (GGA) and local density approximation (LDA) func-
tionals), 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 ad-
equacy of the DFT+Umethod to study TiO2polymorphs. The
DFT+Umethod has been recently utilized to investigate elec-
tron 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
0021-9606/2011/135(5)/054503/9/$30.00 © 2011 American Institute of Physics135, 054503-1
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054503-2 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
Anatase [100] Columbite [001]Rutile [001]
FIG. 1. Schematic crystal structure of the TiO2polymorphs under investigation; anatase, rutile, and columbite.
Heyd-Scuseria-Ernzerhof functional (HSE06) (Refs. 24–26)
recently implemented in the Vienna ab initio simulation pack-
age (VA SP ).
Figure 1shows the crystal structure of the three
polymorphs considered in this work, rutile, anatase, and
columbite, which are all based on octahedral TiO6units. In
the rutile structure (hcp packing of oxygen atoms), each tita-
nium octahedral shares two opposite edges along the caxis
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 aand baxis. The
anatase and rutile varieties can be formed at ambient pres-
sure. Early calorimetry works already determined the ru-
tile 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 mor-
phology affecting the thermodynamic stability of the TiO2
polymorphs has been investigated.28–30 It is nowadays rec-
ognized that anatase is the stable nanophase of TiO2, that
is, anatase gets thermodynamically stabilized by small par-
ticle size.28 Among the TiO2polymorphs stabilized under
high pressure conditions (columbite, baddeleyite, and cotun-
nite) we choose the columbite form (orthorhombic α-PbO2),
in which oxygen atoms assume a distorted hexagonal close
packing configuration. The TiO6octahedra form edge-sharing
chains parallel to the caxis. Individual chains are connected
by corner-sharing to form a three-dimensional framework.
High pressure-high temperature treatment of either anatase or
rutile TiO2yields the columbite modification, which can be
quenched to ambient conditions.31–33
One problem encountered using the DFT+Umethod
is the determination of an appropriate Uparameter value
for each compound. The values of Ucan be determined
through a linear response method which is fully consistent
with the definition of the DFT+UHamiltonian, 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 +Uworks found a
value of U=10 eV for rutile TiO2by fitting calculated
band gaps to the experimental value.15,21 In this work we
discuss the suitability of the DFT+Umethod utilizing the
GGA+Uand the LDA+Ufunctionals to investigate TiO2
polymorphs and how the Uparameter (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+Umethod and the hybrid HSE06 functional
are suitable to investigate crystal structure and electrical
properties of TiO2polymorphs, improving the accuracy
of conventional DFT for band gaps and lattice parameters
prediction. Further, while the conventional DFT, the hybrid
functionals, and the LDA+Ufail to reproduce the phase
stability at both ambient and high pressure, proper results
are obtained within the GGA+Umethod. To the best of
our knowledge, it is the first time that the suitability of the
DFT+Uto 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 exchange-
correlation 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 uti-
lizing 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 exchange-
correlation functionals have been used together with their
LDA+Uand GGA+Uvariants as implemented in VASP.
The Ti(3p,3d,4s) and O (2s,2p) were treated as valence
states. Test calculations performed including the Ti-3sas va-
lence states yielded equivalent results. DFT+Ucalculations
were performed following the simplified rotationally invariant
form proposed by Dudarev.38,42 Within this approach, the on-
site Coulomb term Uand the exchange term J, can be grouped
together into a single effective parameter (U-J) and this effec-
tive parameter will be simply referred to as Uin this paper.
The Jvalue was fix to 1 eV. Effective Uvalues of U=3,
5, 8, and 10 eV were used for the Ti-3dstates. The energy
cut off for the plane wave basis set was kept fix at a con-
stant value of 600 eV throughout the calculations. The recip-
rocal space sampling was done with k-point Monckhorst-Pack
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054503-3 DFT+
U
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
gridsof6×6×8 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 re-
calculated so as to correct the changes in the basis set of the
wave functions during relaxation. Second, from the relaxed
structure, within the GGA+Uapproximation, calculations
were performed at various constant volumes and the energy-
volume data were fitted to the Birch-Murnaghan equation of
state:43
E(V)=E0+9V0B0
16 B0[x2/3−1]3
+[x2/3−1]2[6 −4x2/3],(1)
where x=(V0/V), being Vand V0the volume at pressure pand
the equilibrium volume at ambient pressure, respectively. B0
and B
0are the bulk modulus at ambient pressure and its pres-
sure derivative, respectively, and E0is the equilibrium energy.
Hybrid calculations were performed using the HSE06
functional, as implemented in VASP, with a screening param-
eter μ=0.2 Å−1.The NKRED parameter was set to 2 mean-
ing 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 cal-
culations (the NKRED parameter was omitted, and a normal
grid was utilized in the HF calculation).
III. RESULTS AND DISCUSSION
A. Crystal structure
Table Icompares the calculated lattice parameters and
volume for the fully relaxed structures of TiO2polymorphs
(HSE06, GGA/LDA , and GGA+U/LDA+Uwith 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)forTiO
2polymorphs obtained after complete structural optimization with HSE06,
conventional GGA/LDA, and GGA+U/LDA+Ucompared to experiments. Numbers in parentheses indicate the percent of deviation from experimental data.
Polymorph Method U(eV) V(Å3) a(Å) b(Å) c(Å) (c/a)
ANATASE HSE06 34.07 (0) 3.766 (–0.50) 9.609 (1.02) 2.551 (1.51)
GGA 35.13 (3.11) 3.807 (0.58) 9.693 (1.90) 2.545 (1.27)
GGA+U3 36.15 (6.10) 3.853 (1.79) 9.742 (2.42) 2.529 (0.64)
GGA+U5 36.81 (8.04) 3.881 (2.53) 9.774 (2.76) 2.518 (0.20)
GGA+U8 37.81 (10.9) 3.922 (3.62) 9.830 (3.34) 2.506 (–0.28)
GGA+U10 38.46 (12.9) 3.947 (4.28) 9.871 (3.77) 2.501 (–0.48)
LDA 33.29 (–2.28) 3.74 (–1.19) 9.521 (0.09) 2.546 (1.31)
LDA+U3 34.24 (0.50) 3.793 (0.21) 9.52 (0.08) 2.51 (–0.12)
LDA+U5 34.84 (2.26) 3.819 (0.89) 9.555 (0.45) 2.502 (–0.44)
LDA+U8 35.82 (5.14) 3.86 (1.98) 9.617 (1.10) 2.491 (–0.87)
LDA+U10 36.50 (7.13) 3.882 (2.56) 9.687 (1.84) 2.495 (–0.72)
Experiment31 34.07 3.78512(8) 9.51185(13) 2.513
RUTILE HSE06 31.08 (–0.45) 4.590 (–0.08) 2.950 (–0.29) 0.642 (–0.31)
GGA 32.10 (2.82) 4.650 (1.22) 2.968 (0.32) 0.638 (–0.93)
GGA+U3 32.87 (5.28) 4.671 (1.68) 3.012 (1.80) 0.645 (0.16)
GGA+U5 33.41 (7.01) 4.687 (2.03) 3.042 (2.82) 0.649 (0.78)
GGA+U8 34.17 (9.45) 4.709 (2.51) 3.081 (4.13) 0.654 (1.55)
GGA+U10 34.69 (11.1) 4.725 (2.86) 3.108 (5.05) 0.658 (2.17)
LDA 30.32 (–2.88) 4.556 (–0.82) 2.922 (–1.24) 0.641 (–0.46)
LDA+U3 31.12 (–0.32) 4.580 (–0.30) 2.967 (0.28) 0.648 (0.62)
LDA+U5 31.68 (1.47) 4.599 (0.11) 2.995 (1.23) 0.651 (1.09)
LDA+U8 32.33 (3.55) 4.619 (0.55) 3.031 (2.45) 0.656 (1.86)
LDA+U10 32.92 (5.44) 4.637 (0.94) 3.062 (3.49) 0.66 (2.48)
Experiment (Refs. 31 and 63) 31.22 4.5938(1) 2.9586(1) 0.644
COLUMBITE HSE06 30.52 (–0.23) 4.539 (–0.04) 5.495 (0.04) 4.893 (–0.26) 1.077 (–0.28)
GGA 31.51 (3.00) 4.579 (0.84) 5.576 (1.51) 4.936 (0.61) 1.078 (–0.18)
GGA+U3 32.23 (5.36) 4.607 (1.45) 5.585 (1.67) 5.008 (2.08) 1.087 (0.65)
GGA+U5 32.77 (7.13) 4.629 (1.94) 5.603 (2.00) 5.053 (3.00) 1.092 (1.11)
GGA+U8 33.56 (9.71) 4.658 (2.58) 5.635 (2.59) 5.115 (4.26) 1.098 (1.67)
GGA+U10 34.09 (11.4) 4.674 (2.93) 5.664 (3.11) 5.151 (4.99) 1.102 (2.04)
LDA 29.78 (–2.65) 4.506 (–0.77) 5.450 (–0.78) 4.850 (–1.14) 1.076 (–0.37)
LDA+U3 30.53 (–0.20) 4.526 (–0.33) 5.477 (–0.33) 4.925 (0.38) 1.088 (0.74)
LDA+U5 31.05 (1.50) 4.542 (0.02) 5.501 (0.02) 4.971 (1.32) 1.094 (1.30)
LDA+U8 31.83 (4.05) 4.572 (0.68) 5.537 (0.68) 5.030 (2.53) 1.1 (1.85)
LDA+U10 32.35 (5.75) 4.591 (1.10) 5.563 (1.10) 5.066 (3.26) 1.103 (2.13)
Experiment (Ref. 31.) 30.59 4.541(6) 5.493(8) 4.906(9) 1.080
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054503-4 Arroyo, Morales, and Taravillo 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+Uas a function of U, reaching a maximum
difference at U=10 eV of around 4%. Within the conven-
tional LDA the unit cell volume and lattice parameters are
underestimated. The effect of the Hubbard Uis to expand the
unit cell, and a good reproduction of the experimental crystal
cell is achieved at Uvalues below 5 eV. The HSE06 underes-
timates 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 re-
produce the crystal structure of TiO2polymorphs 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 Uparameter in the
crystal structure prediction, it is more effective to analyze lat-
tice 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 2plots the variation of the c/a ratio for the
different polymorphs versus the Hubbard-U. The experimen-
tal values are indicated by a horizontal line. HSE06 functional
-101234567891011
1.075
1.080
1.085
1.090
1.095
1.100
1.105
Experimental
Experimental
Rutile
Anatase
Columbite
Ex perimental
U (eV)
0.635
0.640
0.645
0.650
0.655
0.660 GG A
HSE0 6
LD A
2.49
2.50
2.51
2.52
2.53
2.54
2.55
-101234567891011
Calculated c/a
FIG. 2. Calculated DFT+Uvalues for the c/a ratio for the optimized crystal
structures of TiO2polymorphs as a function of the Uparameter (for con-
ventional DFT U=0eV);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.
values are denoted by stars. For rutile and columbite, where
the TiO6octahedra share edges along the caxis, 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. 2that in-
dependent 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+Ufor columbite and
anatase small Uvalues of 0.7 and 2.6 eV, respectively, match
the experimental value. The required Uvalues 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 poly-
morph, HSE06 and GGA/LDA show larger deviations to the
experimental c/a ratio, and Uvalues of 6 eV (GGA+U) and
3eV(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 prin-
ciple methods to reproduce this structure, compared to the ru-
tile for which lower deviations to experimental values have
been observed.17,19 Worth mentioning, while adequate Uval-
ues improve the results obtained within the conventional DFT,
very large Uvalues lead to a poor crystal lattice prediction.
Note the large deviation for the columbite polymorph when U
>6eV.
B. Electronic structure of anatase and rutile
One of the most common approaches to determine the
appropriate value of Uis to compare the calculated band gaps
for a set of Uvalues 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 Uvalue, we discuss the general fea-
tures 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 3shows the band gap values extracted in this work
0246810
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
0246810
Anatase
Rutile
Rutile
Anatase
Anatase GGA
Rutile GGA
Experimental
HSE06
Anatase LDA
Rutile LDA
Calculated Band Gap (eV)
U (eV)
FIG. 3. Calculated DFT+Uband-gap for anatase (squares) and rutile (tri-
angles) TiO2polymorphs as a function of the Uparameter; LDA+Uhollow
symbols and GGA+Ufilled symbols. Lines are a guide to eye. Stars corre-
spond to the HSE06 values. Experimental data are indicated by horizontal red
bars.
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054503-5 DFT+
U
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
dxy
dxz dyz
M- O σ*
M- O π*
M- M
M- O π
M- O σ
-6
-4
-2
0
2
4
6
8
Calculated DOS
t2g
eg
Energy (eV)
TOTAL
Ti (d)
O (p)
O (p)
Ti (eg)
Ti (t2g)
II c
FIG. 4. Calculated density of states (DOS) of rutile-TiO2within 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-TiO2band 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 con-
ventional DFT (rutile ∼1.86 eV, anatase ∼2.07 eV) are
similar to those obtained in previous DFT studies.19,20 Hy-
brid 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 orig-
inal 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+Uthe band gap increases
with the value of the Uparameter, confirming the fact that a
Hubbard-like correction term (U) substantially improves the
accuracy of the calculated band-gap, compared to the con-
ventional DFT. In good agreement with other authors,15,21
values of 10 and 8.5 eV are required to match the experi-
mental band gap of rutile and anatase, respectively. However,
these values seem too large if one takes into account Uval-
ues 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 fea-
tures in the electronic structure than the band gap should be
examined.
Figure 4shows the total calculated DOS of rutile to-
gether with the partial DOS of Ti and O in green and red,
respectively. The Fermi level is arbitrarily chosen as the ori-
gin 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 oc-
cupied dorbitals, and therefore for a purely ionic Ti–O in-
teraction no Ti-dstates should be occupied. This means that
any dcharacter in the DOS below the Fermi level is a direct
result of oxygen–titanium covalent interactions. The σover-
lap between Ti-3d(eg) and O-2porbitals results in the bonding
σband, which appears below the Fermi level (predominantly
oxygen-2pin character), and the σ*band above the Fermi
level (mostly consisting of titanium-3dstates). The significant
mixing of O-2pstates and Ti-3dstates in both the σand σ*
bands supplies direct evidence for the strong covalent interac-
tion of the Ti–O bonds. The titanium t2gorbitals (dxy,dxz, and
dyz) are involved in πinteractions with the filled oxygen por-
bitals 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 t2gor-
bitals are degenerate. In rutile, however, the TiO6octahedra
share edges along the caxis, and vertex in the ab plane. This
distortion breaks the degeneration of the t2gorbitals. 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 5shows the evolution of the DOS with the Upa-
rameter within the GGA+U. A similar evolution is observed
within the LDA+U(see Ref. 51). Increasing the Uvalue has
two effects: (i) opening the energy gap between the valence
and conduction bands and (ii) a progressive merging of the t2g
and egderived bands, which do not differentiate any longer
at U=10 eV. The Uparameter keeps the dorbitals atomic
like, diminishing the effective overlapping of O-2pand Ti-3d
orbitals. In rutile, the downshift of the σ*band is related to
a reduction in the Ti(eg)-Opoverlapping (weaker Ti–O bond,
longer alattice parameter), while the up shift and narrowing
of the t2g-derived band is associated to a weaker Ti–Ti inter-
action (longer clattice parameter). Changes of the t2gband as
afunctionofUare more pronounced that those of the egband
as the clattice parameter increases more than the aparame-
ter (c/a ratio increases). Therefore, the DOS modifications as
a function of the Uparameter are consistent with the evolu-
tion of the lattice parameters and the c/a ratio (see Table Iand
Fig. 2).
To further analyze the adequacy of the DFT+Umethod,
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
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054503-6 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
-6 -4 - 2 0 2 4 6 8
Energy (eV)
HSE06
GGA
U = 8 eV
U = 10 eV
U = 3 eV
U = 5 eV
Calculated DOS of
Rutile
Ti O2
-6 -4 - 2 0 2 4 6 8
CFS = 2.4 eV
CFS = 1.9 eV
CFS = 1.6 eV
CFS = 1.2 eV
CFS = 0.9 eV
CFS = 2.6 eV
FIG. 5. DOS of rutile TiO2calculated using the HSE06, the conventional
GGA method, and the GGA+Uwith Uvalues of 3, 5, 8, and 10 eV. Guide-
lines show the displacement of the bands as the value of Uincreases. CFS
refers to the crystal field splitting between averaged energies of t2gand eg
derived bands. The DOS calculated using the LDA+Uis shown in the sup-
plementary information.
energy difference within the average energies of the t2g
and egbands (crystal field splitting, CFS) varies from 1.8
to 3.0 eV.53,54 It can be seen in Fig. 5that a too large
Uvalue (U>5 eV) dramatically reduces the CFS (for
LDA+Uresults see Ref. 51). Similarly, for anatase-TiO2
(experimental CFS value 2.6 eV (Ref. 54)), the introduc-
tion of the Uparameter in the calculation reduces the CFS
value; 2.2 eV for GGA, 1.7 eV for GGA+U,U=5eV
(see Ref. 51). In short, the DFT+Umethod allows a good
qualitative description of the electronic structure for moder-
ate Uvalues, below 5 eV. Using these results as a starting
point, the Uvalue 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 TiO2at 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 Hy-
brid functionals also produce wrong phase stability, regard-
less the amount of HF exchange included.19 Figure 6repre-
sents 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
0246810
-0.09
-0.06
-0.03
0.00
0.03
0.06
0.09
0.12
0.15
0.18
Rao 1961
Levchenko 2006
JANAF 1973
Ranade 2002
Calculated energy difference (eV/f.u.)
U (eV)
Anatase GGA
Columbite GGA
Anatase LDA
Columbite LDA
Anatase HSE06
Columbite HSE06
Exp. Anatase
Less stable than Rutile
GGA+U
Erut< E anat < E col
FIG. 6. Calculated DFT+Utotal energy differences for TiO2polymorphs as
a function of the Uparameter; LDA+Uhollow symbols and GGA+Ufilled
symbols. Data are referred to the rutile polymorph. Eanatase–Erutile is denoted
by squares (DFT+U) and star (HSE06). Erutile–Ecolumbite is denoted by tri-
angles (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 Uvalues for what the GGA+Ure-
produces the experimental phase stability.
functional. Positive energy differences indicate that a poly-
morph 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 ru-
tile 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 Uparameter on phase
stability. It can be seen in Fig. 6that the relative energy of the
polymorphs strongly depends on the value of U. Noteworthy,
the LDA and GGA functionals predict different energetic sta-
bility. Within LDA+Uthe 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 ob-
served 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+Uin 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 (Erutile <Ecolumbite <Eanatase).
To investigate phase stability under pressure, several
fixed-volume calculations were performed starting from the
GGA and GGA+Uoptimized structures. The strong failure
of the LDA and hybrid functionals to reproduce phase sta-
bility at ambient pressure, at the expense of a large compu-
tational effort for the latter, discourages the investigation un-
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054503-7 DFT+
U
calculations of TiO2polymorphs J. Chem. Phys. 135, 054503 (2011)
26 28 30 32 34 36 38 40 42
-27.0
-26.8
-26.6
-26.4
-26.2
-26.0
-25.8
26 28 30 32 34 36 38 40 42
-24.8
-24.6
-24.4
-24.2
-24.0
-23.8
-23.6
26 28 30 32 34 36 38 40 42
-23.4
-23.2
-23.0
-22.8
-22.6
-22.4
26 28 30 32 34 36 38 40 42
-20.4
-20.2
-20.0
-19.8
-19.6
-19.4
ANATASE RUTILE COLUMBITE
Volume ( Å
3
/f.u.)
Calculated Energy (eV/f.u.)
DFT
U = 5 eV U = 10 eV
U = 3 eV
FIG. 7. Total energy vs. volume curves for TiO2polymorphs. Symbols correspond to the conventional GGA and GGA+Ucalculated data, and lines show the
fitting to the Birch-Murnaghan equation of state.
der pressure. Figure 7shows 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 energy-
volume data for the investigated TiO2forms. The bulk mod-
uli are clearly underestimated, independent of the choice of
U. Note the major differences with experimental bulk mod-
uli are observed for U=10 eV. For anatase and rutile in-
creasing Uvalues lead to lower bulk moduli, a trend observed
in other systems: Lu2O3,56 FeS2,57 and MnO.8The fluctua-
tions of the bulk moduli as a function of Uobtained for the
Columbite phase are not surprising neither (see, for instance,
Ce2O3(Ref. 58)orFe
1–xMgxO(Ref.59)).
In Fig. 7it 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 be-
ing a metastable phase at any pressure. The GGA+Ucorrects
TABLE II. Calculated equation of state parameters for TiO2polymorphs (conventional GGA vs. GGA+U). E0,
V0,B0,andB
0are the zero-pressure energy, volume, bulk modulus, and its pressure derivative, respectively.
Polymorph Method U(eV) E0(eV/f.u.) V0(Å3/f.u.) B0(GPa) B
0rms (eV/f.u.)
ANATASE GGA –26.9045 35.272 169.9 2.27 0.2
GGA+U3 –24.7312 36.239 164.9 2.53 0.2
GGA+U5 –23.3733 36.902 162.3 2.61 0.3
GG+U10 –20.2905 38.572 157.6 2.67 0.4
Experiment (Ref. 31.) 34.07 179 ±24.5
RUTILE GGA –26.8128 32.186 200.4 4.98 0.7
GGA+U3 –24.6944 32.946 199.5 4.77 0.6
GGA+U5 –23.3727 33.458 198.7 4.62 0.6
GGA+U10 –20.3735 34.740 195.3 4.40 0.5
Experiment (Ref. 63.) 31.22 211 ±76.76
COLUMBITE GGA –26.8288 31.651 195.9 4.22 1.1
GGA+U3 –24.6718 32.334 206.8 3.62 0.3
GGA+U5 –23.3363 32.866 202.3 3.69 0.4
GGA+U10 –20.3171 34.179 193.9 3.75 0.3
Experiment (Ref. 31.) 30.59 258 ±84.1
rms =(E−Efit)2
n.
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054503-8 Arroyo, Morales, and Taravillo J. Chem. Phys. 135, 054503 (2011)
-0.04
-0.02
0.00
0.02
0.04
0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0 2 4 6 8 1012141618
0 2 4 6 8 1012141618
0 2 4 6 8 1012141618
-0.04
0.00
0.04
0.08
0.12
U = 3 eV
Prut-col
tr = 6.1 GPa
Pana-col
tr = 2.5 GPa
Calculated Enthalpy difference (eV/f.u.)
Pressure (GPa)
Pana-rut
tr = 1.8 GPa
U = 5 eV
Prut-col
tr = 10.1 GPa
Pana-col
tr = 1.5 GPa
Pana-rut
tr = 0.03 GPa
U = 10 eV
Prut-col
tr = 16 GPa
Rutile
Anatase
Columbite
FIG. 8. Calculated Enthalpy vs. pressure for TiO2polymorphs within the
GGA+U(U=3, 5, and 10 eV).
these discrepancies with experiments. For U=3eVtheru-
tile 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>5eV)ata
certain volume, indicating that columbite becomes more sta-
ble at a sufficient pressure. Comparing the predicted pressure
of the anatase →columbite and rutile →columbite transfor-
mations with experimental data is another way to determine
appropriate Uvalues.
D. Transition pressures
Figure 8shows the calculated enthalpy-pressure variation
for the TiO2polymorphs (at 0 K), within the GGA+U. The ef-
fect of Uis to extend the stability field of the rutile phase, rais-
ing/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 parti-
cle size. Rutile undergoes the transformation to columbite at
about 10 GPa.33,60 Therefore, the GGA+Uwith U=5eV
provides the best agreement with experiments (PA-C =1.5
GPa, PR-C =10 GPa). At U=10 eV the rutile transforma-
tion occurs at too high pressure (16 GPa), and the anatase to
columbite transformation is not feasible.
IV. CONCLUSIONS
The suitability of DFT+Umethodology to investigate
TiO2polymorphs is discussed and compared to conven-
tional DFT and hybrid HSE06 functionals. The GGA+U
and LDA+Umethods 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 Uparameter to repro-
duce all TiO2properties. Reproducing the experimental band
gap of rutile-TiO2requires a Uparameter of 10 eV, which in
turn produces a band structure in disagreement with experi-
ments. In addition, such large Uvalues worsen the reproduc-
tion of crystal structure. Generally speaking, we found that
the DFT+U(U≈5 eV) method is well suited to investigate
the properties of individual TiO2polymorphs (anatase, rutile,
and columbite). Yet, the hybrid HSE06 performs equally well
to accurately reproduce the crystal and electronic structures
of TiO2polymorphs.
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 Hub-
bard correction term (U) in the conventional LDA (LDA+U)
has the effect of excessively stabilizing the columbite poly-
morph, which is predicted as more stable than the anatase
form at any Uvalue. Note that contrary to any experimen-
tal observation columbite is even more stable than rutile for
U<2.6 e V. The GGA+Umethod yields an ener-
getic sequence consistent with experiments, Erutile <Eanatase
<Ecolumbite,forUvalues 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 Uvalues 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 phase-
stability of TiO2.
We conclude that computational results for titania poly-
morphs 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 par-
ticular, 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. S2009-
PPQ/1551 funded by Comunidad de Madrid. A.M.-G. ac-
knowledges 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.
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