Hybrid functional studies of the oxygen vacancy in TiO
2
A. Janotti,
1
J. B. Varley,
1
P. Rinke,
1
N. Umezawa,
1,
* G. Kresse,
2
and C. G. Van de Walle
1
1
Materials Department, University of California, Santa Barbara, California 93106-5050, USA
2
Faculty of Physics, Center for Computational Materials Science, University of Vienna, A-1090 Wien, Austria
Received 4 May 2009; revised manuscript received 31 December 2009; published 16 February 2010
The electronic and structural properties of the oxygen vacancy V
O
in rutile TiO
2
are studied using gener-
alized Kohn-Sham theory with the Heyd, Scuseria, and Ernzerhof HSE hybrid functional for exchange and
correlation. The HSE approach corrects the band gap and allows for a proper description of defects with energy
levels close to the conduction band. According to the HSE calculations, V
O
is a shallow donor for which the +2
charge state is lower in energy than the neutral and +1 charge states for all Fermi-level positions in the band
gap. The formation energy of V
O
2+
is relatively low in n-type TiO
2
under O-poor conditions but it rapidly
increases with the oxygen chemical potential. This is consistent with experimental observations where the
electrical conductivity decreases with oxygen partial pressure.
DOI: 10.1103/PhysRevB.81.085212 PACS numbers: 71.55.-i, 61.72.Bb
I. INTRODUCTION
TiO
2
is a material of increasing interest in electronics and
optoelectronics, with applications in high-k dielectrics, solar
cells, and photocatalysis.
1–5
Rutile TiO
2
has a wide band gap
of 3.1 eV and exhibits a tendency for unintentional n-type
conductivity.
6–12
Understanding and controlling this conduc-
tivity would be a key step toward the application of TiO
2
as
a semiconductor. It has been widely reported that the n-type
conductivity in TiO
2
varies inversely with the oxygen partial
pressure in the annealing atmosphere.
7–12
These observations
have led to the conclusion that oxygen vacancies are the
cause of the n-type conductivity.
4,9–13
It has also been argued
that TiO
2
can be easily reduced, and that it supports a high
degree of nonstoichiometry in the form of oxygen vacancies
TiO
2-x
.
4,13–15
Despite the many years of research on TiO
2
,
direct evidence of the role of oxygen vacancies in the n-type
conductivity is still lacking, and a microscopic understanding
of the electronic and structural properties of the oxygen va-
cancy has remained elusive. Most experiments have focused
on the surface of rutile TiO
2
, and it is not clear whether
annealing under oxygen-poor or oxygen-rich atmospheres
truly affects the material as a whole or only a thin surface
layer. Unintentional incorporation of dopant impurities, such
as hydrogen, may further complicate the interpretation of
electrical measurements. In addition, for ZnO both experi-
ment and theory have recently shown that, contrary to con-
ventional wisdom, the oxygen vacancy is a deep rather than
a shallow donor.
16–18
It is therefore opportune to revisit the
role of oxygen vacancy V
O
in bulk TiO
2
.
Density-functional theory DFT has become the method
of choice for studying the electronic structure of isolated
point defects in semiconductors and insulators.
19
The most
common exchange-correlation functionals in this context are
the local-density approximation LDA or the generalized
gradient approximation GGA. However, the limitations of
LDA and GGA in predicting band gaps pose serious prob-
lems to the description of the electronic and structural prop-
erties of oxygen vacancies in TiO
2
.
20,21
The removal of an
oxygen atom from the TiO
2
lattice results in a doubly occu-
pied a
1
single-particle state in the band gap, that is very near
the conduction-band minimum CBM in the LDA or GGA.
When lattice relaxations are included, this a
1
state moves up
in energy and merges with the conduction band. The two
electrons are thus transferred to the CBM, rendering it im-
possible to stabilize a neutral or +1 charge state of the va-
cancy in which these electrons reside in localized states on
the vacancy. Therefore, in the LDA or GGA, V
O
behaves as
a shallow donor. The question is whether this reflects the true
physics of the V
O
center or rather an artifact of LDA or GGA
due to the underestimation of the band gap.
It is therefore necessary to use methods that overcome the
band-gap problem in order to correctly describe the elec-
tronic and structural properties of the oxygen vacancy in
TiO
2
. Here we investigate the oxygen vacancy in rutile TiO
2
using the Heyd, Scuseria, and Ernzerhof HSE hybrid
functional,
22
in which a portion of Hartree-Fock HF ex-
change is range limited and mixed with GGA exchange and
correlation. HSE not only corrects the band gap of TiO
2
see,
e.g., Fig. 1d but also stabilizes the neutral state in the band
gap and thus provides the opportunity to correlate the posi-
tion of the single-particle a
1
state with the local lattice relax-
ations. This allows us to compute formation energies and
transition levels, and disentangle the effects of structural re-
laxations in the various charge states of V
O
. The results show
that the oxygen vacancy does act as a shallow donor.
II. COMPUTATIONAL APPROACH
The calculations are based on generalized Kohn-Sham
theory
23
and the projector-augmented wave potentials
24,25
as
implemented in the VASP code.
26,27
For Ti the 3p,3d, and 4s
states were treated as valence states and the Perdew, Burke,
and Ernzerhof PBE potential with a core radius of 2.5 a.u
was used whereas for O the standard PBE potential with a
core radius of 1.5 a.u. was applied. The calculations were
performed using both the GGA of PBE Ref. 28 and the
hybrid functional as proposed by HSE.
22
In the latter, the
exchange potential is separated into a long-range and a short-
range part, and HF exchange is mixed with PBE exchange
only in the short-range part. The long-range part of the ex-
change potential is therefore essentially described by PBE.
PHYSICAL REVIEW B 81, 085212 2010
1098-0121/2010/818/0852127 ©2010 The American Physical Society 085212-1