First-principles study of intrinsic point defects in ZnO:
Role of band structure, volume relaxation, and finite-size effects
Paul Erhart, Karsten Albe, and Andreas Klein
Institut für Materialwissenschaft, Technische Universität Darmstadt, Petersenstraße 23, D-64287 Darmstadt, Germany
Received 25 November 2005; revised manuscript received 6 February 2006; published 15 May 2006
Density-functional theory DFT calculations of intrinsic point defect properties in zinc oxide were per-
formed in order to remedy the influence of finite-size effects and the improper description of the band structure.
The generalized gradient approximation GGA with empirical self-interaction corrections GGA+ U was
applied to correct for the overestimation of covalency intrinsic to GGA-DFT calculations. Elastic as well as
electrostatic image interactions were accounted for by application of extensive finite-size scaling and compen-
sating charge corrections. Size-corrected formation enthalpies and volumes as well as their charge state de-
pendence have been deduced. Our results partly confirm earlier calculations but reveal a larger number of
transition levels: 1 For both the zinc interstitial as well as the oxygen vacancy, transition levels are close to
the conduction band minimum. 2 The zinc vacancy shows a transition rather close to the valence band
maximum and another one near the middle of the calculated band gap. 3 For the oxygen interstitials,
transition levels occur both near the valence band maximum and the conduction band minimum.
DOI: 10.1103/PhysRevB.73.205203 PACS numbers: 61.72.Bb, 61.72.Ji, 71.15.Mb, 71.55.Gs
I. INTRODUCTION
The current interest in zinc oxide is largely driven by
potential applications in optical and optoelectronic devices.
1
Since many properties of zinc oxide are highly sensitive to
point and line defects present in the material, the defect
physics of ZnO has been extensively studied in the past.
Theoretically, a number of density functional theory DFT
calculations have been performed to elucidate the behavior
of intrinsic
2–5
as well as extrinsic point defects.
6–9
These cal-
culations, however, are based on the local density LDA or
generalized-gradient approximation GGA which suffer
from an underestimation of the band gap and an improper
description of the band structure. The first shortcoming is
intrinsic to the DFT method in general see, e.g., Refs. 10
and 11. The second problem is particularly pronounced for
zinc oxide because self-interactions intrinsic to the LDA and
GGA exchange-correlation potentials cause an energy level
shift of the Zn 3d states. As a result, the calculations not only
yield a band-gap error of more than 2 eV but also overesti-
mate the covalency of the Zn-O bond. A direct comparison
between data calculated within LDA or GGA-DFT and ex-
periment is, therefore, severely hampered. In the past, this
problem has been addressed in various ways.
Zhang et al. proposed an empirical correction scheme
based on a Taylor expansion of the formation enthalpies in
the plane-wave cutoff energy.
3,12
Since a profound physical
motivation for this scheme is lacking, the results can only be
interpreted semiquantitatively. Kohan et al. discussed correc-
tions based on the electronic structure of the defect
configurations,
2
while other authors resorted to a qualitative
discussion of their results.
4,5
If no correction is applied the calculated formation enthal-
pies reported by different authors are comparable see Table
I below, whereas the various correction schemes lead to
very different results. This can be illustrated for the case of
the oxygen vacancy. According to the data of Kohan et al.
the +2 / 0 transition for this defect should be located in the
vicinity of the valence band maximum VBM,
2
while the
corrected data by Zhang et al. predict the same transition to
occur just below the conduction band minimum CBM.
3
Since it is difficult to assess the reliability of these predic-
tions, quantitatively more reliable calculations are required.
Recently, some defect calculations were carried out using
the semiempirical LDA+ U scheme,
13
which allows one to
adjust the position of d-electron levels by implementing self-
interaction corrections into the LDA or GGA exchange-
correlation potentials. Hitherto, this scheme has been em-
ployed to study point defects in CuInSe
2
, where the Cu 3d
electrons play a similar role as the Zn 3d electrons in ZnO,
14
and in calculations of optical transition levels of the oxygen
vacancy in ZnO.
15,16
Therefore the method is an excellent
candidate for a reassessment of the thermodynamics of point
defects in zinc oxide. Another issue, which has hardly been
addressed in studies of point defects in zinc oxide so far, is
the role of volume relaxation and finite-size effects. It is,
however, well-known, that formation enthalpies can con-
verge slowly with supercell size,
17
especially if charged de-
fects are considered.
18
The purpose of the present work is twofold. First, we seek
to determine formation enthalpies for the intrinsic point de-
fects of zinc oxide by taking into account the role of Zn 3d
electrons. Furthermore, we study the effect of supercell size
and volume relaxation by employing finite-size scaling.
Thereby, we are also able to obtain defect formation volumes
for point defects in ZnO. In summary, by taking into account
the band structure as well as finite-size effects, this study
aims to provide a consistent set of point defect properties,
which will allow for a more quantitative interpretation of
experimental data.
In the following section, we summarize some observa-
tions on the band structure of zinc oxide based on experi-
mental as well as theoretical studies. This overview allows us
to motivate our computational approach which is described
PHYSICAL REVIEW B 73, 205203 2006
1098-0121/2006/7320/2052039 ©2006 The American Physical Society 205203-1
Urheberrechtlich geschützt / In Copyright