IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 195503 (6pp) doi:10.1088/0953-8984/24/19/195503

The electronic structure and opticalresponse of rutile, anatase and brookiteTiO2

M Landmann, E Rauls and W G Schmidt

Lehrstuhl fur Theoretische Physik, Universitat Paderborn, D-33095 Paderborn, Germany

E-mail: landmann@mail.upb.de

Received 4 January 2012, in final form 21 March 2012Published 19 April 2012Online at stacks.iop.org/JPhysCM/24/195503

AbstractIn this study, we present a combined density functional theory and many-body perturbationtheory study on the electronic and optical properties of TiO2 brookite as well as the tetragonalphases rutile and anatase. The electronic structure and linear optical response have beencalculated from the Kohn–Sham band structure applying (semi)local as well as nonlocalscreened hybrid exchange–correlation density functionals. Single-particle excitations aretreated within the GW approximation for independent quasiparticles. For optical responsecalculations, two-particle excitations have been included by solving the Bethe–Salpeterequation for Coulomb correlated electron–hole pairs. On this methodological basis, gap dataand optical spectra for the three major phases of TiO2 are provided. The commoncharacteristics of brookite with the rutile and anatase phases, which have been discussed morecomprehensively in the literature, are highlighted. Furthermore, the comparison of the presentcalculations with measured optical response data of rutile indicate that discrepancies discussedin numerous earlier studies are due to the measurements rather than related to an insufficienttheoretical description.

(Some figures may appear in colour only in the online journal)

1. Introduction

Of the manifold of synthesizable crystalline TiO2 poly-morphs, mainly the three crystal types rutile {crystalsystem: tetragonal (4/mmm), space group: P42/mnm–D14

4h},anatase {tetragonal (4/mmm), I41/amd–D19

4h} and brookite{orthorhombic (mmm), Pbca–D15

2h} are found in nature. Onlythe first two play a significant role in industrial applications.Experimental data on TiO2 brookite is limited due to itsrareness and difficult preparation [1].

As a semiconductor material with long-term stability,non-toxic environmental acceptability and broadly low costavailability TiO2 has also been taken into account forphotovoltaic applications. However, due to optical gapsslightly above 3 eV (rutile: ∼3.0 eV [3, 4], anatase:∼3.4 eV [5] and brookite: ∼3.3 eV [6]), natural TiO2 is onlyphotoactive in the UV region of the electromagnetic spectrumand an inefficient active solar cell material. Still, the material

advantages of TiO2 can be used indirectly in technically andeconomically viable dye-sensitized solar cells [7–9] where itacts as an electron-transporting substrate for a chemisorbedphotoactive dye.

The structure of the three main phases is wellcharacterized by the two complementary TixOy building-block representations pictured in figure 1. The most commonrepresentation is the view of the crystal structures as networksof edge- and/or corner-linked distorted TiO6 octahedronbuilding blocks. Rutile and brookite exhibit corner—as wellas edge-sharing TiO6 units. In rutile, the Oh symmetryof an ideal octahedron is reduced to D2h symmetry dueto different in-plane (equatorial) and out-of-plane (axial)Ti–O bond lengths and two types of Ti–O–Ti in-plane bondangles deviating from 90◦. In the anatase phase, additionaldisplacements of the O ions from the equatorial positionsgenerate a local D2h symmetry seen by the Ti ions. Due tostronger distortions in TiO2 brookite, all bond lengths and

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J. Phys.: Condens. Matter 24 (2012) 195503 M Landmann et al

Figure 1. Planar Ti3O building-block representation (left) and TiO6polyhedra (right) for the TiO2 phases rutile (a), anatase (b) andbrookite (c) (Ti (white); O (red)). The experimental latticeparameters a = b = 4.5937 A, c = 2.9581 A for rutile [10],a = b = 3.7842 A, c = 9.5146 A for anatase [11] anda = 9.16 A, b = 5.43 A, c = 5.13 A for brookite [12] have beenused throughout this work.

bond angles slightly differ from each other, thus leading to theformal loss of local symmetry and C1 symmetric TiO6 units.

A complementary view of the crystal structure is givenby the threefold coordination of O ions in trigonal-planar-typeTi3O building blocks. While the Ti3O units exhibit a Y-shapedconformation in rutile, the anatase units are closer to aT-shape. In brookite, both T- and Y-shaped Ti3O units arepresent.

The linear optical absorption properties of rutilehave been investigated, among other studies, in reflectionmeasurements by Cardona and Harbeke [13] and in a morerecent reflection-generalized ellipsometry study by Tiwaldand Schubert [14]. Their results have demonstrated thecharacteristic anisotropy in the optical absorption for E ‖ c(extraordinary polarization) and E ⊥ c (ordinary polarization).However, the reported dielectric functions show significantdeviations. In the Cardona–Harbeke data the spectral weighton the first main peak at ∼4 eV is much more pronounced forE ‖ c than for E ⊥ c. In contrast, more recently published data

by Tiwald and Schubert indicate a less intense peak for E ‖ cat ∼4 eV. Also, the existence of distinct second main peaksabove 6.5 eV in both polarizations, observed in the newerdata, is not resolved in the Cardona–Harbeke data either. Theanisotropies in the optical functions of anatase for E ‖ c andE ⊥ c have been characterized, among others, by Hosakaet al [15]. The optical properties of brookite TiO2 have beeninvestigated in the last few years by several experimentalstudies [17–21, 6], predominantly focusing on the absorptionedge.

Several density functional theory (DFT)-based studieson various TiO2 phases (mainly rutile and anatase) havebeen published so far [22–31]. The improved descriptionof the electronic structure by hybrid-functional schemeshas been discussed in the literature [32–37]. Recently,some studies have investigated many-body effects (self-energy corrections and excitonic effects) on the opticalabsorption [38–40] of rutile and anatase. Lawler et alinvestigated the birefringence of the oxides TiO2 includingexcitonic effects from the solution of the Bethe–Salpeterequation (BSE). Kang and Hybertson [40] as well asChiodo et al [39] both applied many-body perturbationtheory (MBPT) methods (i.e. GW approximation and BSE)to calculate optical properties. Their results have clearlydemonstrated the necessity of quasiparticle corrections toreproduce experimental photoemission bandgap data as wellas the importance of including electron–hole interaction inthe simulation of the optical response of the two tetragonalpolytypes rutile and anatase. So far, a similar investigation forthe more complex brookite phase, whose symmetry-reducedstructure might indicate how optical characteristics in TiO2are affected by increasing disorder, is still missing.

Several authors [39, 40] have attributed the observeddifferences between the theoretical dielectric function and theexperimental data of Cardona and Harbeke [13], among otherreasonings, to possible deficiencies of the BSE formalismin describing all spectral features of the TiO2 rutile phase.However, as we will demonstrate in light of our numericalresults and the comparison with the data published byTiwald and Schubert [14], there is strong evidence for aninsufficient description of the optical properties of rutile bythe Cardona–Harbeke data.

2. Methodology

We have performed DFT calculations [41, 42] usingthe projector-augmented wave (PAW) method [43, 44] asimplemented in the Vienna ab initio simulation package(VASP) [45]. An energy cutoff of 400 eV was used to expandthe Kohn–Sham orbitals into plane wave basis sets. In additionto the 3d and 4s valence states, the 3s and 3p semicore statesof Ti have been explicitly treated as valence states throughoutthis work. We applied both the standard generalized gradientapproximation (GGA) exchange–correlation (XC) functionalaccording to Perdew–Burke–Ernzerhof (PBE) [46] as wellas a modern nonlocal, range-separated, screened Coulombpotential hybrid density functional proposed by Heyd,

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J. Phys.: Condens. Matter 24 (2012) 195503 M Landmann et al

Figure 2. TiO2 band structures for the rutile (left), anatase (middle) and brookite (right) polymorphs. Pictured are the DFT–HSE06 bandstructure as well as QP energies (dots) obtained from PBE–G0W0 calculations. The gray shaded areas correspond to energies below thehighest valence band and above the lowest conduction band from DFT–PBE calculations. For PBE and HSE06 eigenvalues the valence bandmaximum has been chosen as the zero point of the energy scale. G0W0 QP energies are given with respect to DFT–PBE energy scale. TheHSE06 calculations have been carried out with regular 7× 7× 11, 9× 9× 3 and 3× 5× 5 0-centered k-point meshes for rutile anatase andbrookite. The various high symmetry lines were additionally sampled by 40 k-points to obtain the HSE06 band structure. For rutile, anataseand brookite, the GW calculations were carried out on top of DFT–PBE calculations with 192, 384 and 768 electronic bands, regular8× 8× 12, 10× 10× 4 and 4× 6× 6 0-centered k-point meshes, and 192 frequency points for sampling the dielectric function. Thesenumerical parameters provide bandgaps converged within about 0.02 eV accuracy, as tested for the case of rutile.

Scuseria and Ernzerhof (HSE06) [47–50] that corrects, to alarge part, the bandgap underestimation in standard DFT.

Quasiparticle (QP) corrections due to self-energyeffects within the GW approximation have been includedperturbatively in a full-frequency-dependent G0W0 scheme(for details on VASP implementation see [51–54]) on top ofDFT–PBE and DFT–HSE06 calculations.

In order to account for excitonic contributions and localfield effects, the Coulomb correlation of (quasi-)electrons and(quasi-)holes is included by solution of the Bethe–Salpeterequation [55–60] in the Tamm–Dancoff approximation.Instead of an unfavorable O

(N3

)scaling diagonalization of

the electron–hole pair Hamiltonian, the calculation of thefrequency-dependent dielectric function is performed by anO

(N2

)scaling time-evolution scheme proposed by Schmidt

et al [61].

3. Results and discussion

3.1. Electronic properties

The electronic band structures of the three TiO2 modificationsrutile, anatase and brookite are shown in figure 2. BesidesHSE06 and G0W0 results, the DFT–PBE band edge positionsare included to demonstrate the DFT–PBE underestimation ofthe electronic bandgap. The fundamental bandgaps have beensummarized in table 1. The rutile and brookite PBE gaps of1.88 and 1.86 eV are direct (d) at 0. However, the 0-pointconduction band minimum in rutile is almost degenerate withthe local R and minima M (0.03 and 0.05 eV higher inenergy). For anatase, an indirect (id) energy gap of 1.94 eV isobserved between the conduction band minimum at 0 and thevalence band maximum along the 6 path (0.880→ M), that

Table 1. Calculated fundamental bandgaps (given in eV) of theTiO2 main phases.

Method

Fundamental bandgaps

Rutile Anatase Brookite

DFT–PBE 1.88 (0→ 0) 1.94 (0.886→ 0) 1.86 (0→ 0)DFT–HSE06 3.39 (0→ 0) 3.60 (0.896→ 0) 3.30 (0→ 0)PBE–G0W0 3.46 (0→ 0) 3.73 (6→ 0) 3.45 (0→ 0)HSE06–G0W0 3.73 (0→ 0) 4.05 (6→ 0) 3.68 (0→ 0)

is marginally smaller than the direct 0-point gap of 2.07 eV.Using the HSE06 hybrid functional, the bandgaps of rutile,anatase and brookite are increased to 3.39 eV (d), 3.60 eV(id) and 3.30 eV (d), which follows the trends in previoushybrid-functional studies [27, 32, 33, 35, 37].

Our full-frequency-dependent G0W0 calculations predictQP gaps of 3.46 eV (d) for rutile, 3.73 eV (id) for anataseand 3.45 eV (d) for brookite. These results are slightly(0.1–0.15 eV) above the HSE06 values, thus demonstratingthat the QP characteristics of TiO2 are well approximatedby the HSE06 functional. The rutile and anatase bandgapsare comparable to the G0W0 QP gaps of 3.34 eV (0 →

R)/3.38 eV (0 → 0) and 3.56 eV (id) recently reported byKang and Hybertson [40] as well as close to the 3.59 eV (d)and 3.83 eV (id) reported by Chiodo et al [39].

The HSE06 functional reproduces the G0W0 QP bandsclose to the band edges with high accuracy (cf figure 2). Thisconformity becomes significantly worse further away from theband edges as demonstrated by HSE06 O2s states ∼0.7 eVbelow the G0W0 QP energies.

We also tested the influence of different initialwavefunctions by calculating G0W0 QP shifts for HSE06wavefunctions which increases the QP gaps by 0.40± 0.05 eV

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J. Phys.: Condens. Matter 24 (2012) 195503 M Landmann et al

Figure 3. Total and angular momentum decomposed densities of states for the three TiO2 crystal polymorphs rutile, anatase and brookitecalculated with the HSE06 XC functional. Shown are the Ti3d,O2s and O2p contributions to the local densities of states.

(cf table 1). Similar to the HSE06–G0W0 value of 3.73 eV,a bandgap of 3.67 eV is obtained for TiO2 rutile if partialself-consistency in the Green’s function G is included byupdating the eigenvalues four times.

Experimentally, QP properties are probed by direct andinverse photoemission spectroscopy (IPS). For rutile, Tezukaet al [62] reported a bandgap of 3.3 ± 0.5 eV (d) fromcombined photoemission and bremsstrahlung isochromatspectroscopy. In a combined ultraviolet photoemissionspectroscopy (UPS) and IPS study [63], a bandgap of 3.6 eV(d) has been determined. This means that the HSE06 andPBE–G0W0 rutile bandgaps of 3.39 and 3.46 eV agree verywell with the experimental data range. By taking into accounta bandgap of ∼4 eV from angle-resolved photoemissionspectroscopy [23], the HSE06–G0W0 bandgap of 3.73 is alsoa reasonable result. Due to missing photoemission data, QPgaps of ∼3.7 eV for anatase and ∼3.5 eV for brookite couldonly be estimated from the smaller optical gaps by assumingsimilar trends as obtained for rutile. Consequently, the HSE06and PBE–G0W0 calculations provide reliable bandgap data oncrystalline TiO2.

An analysis of the density of states (DOS), depicted infigure 3, shows predominantly O2p-like valence band statesand Ti3d-like conduction band states around the band edgesfor all three major phases. As indicated in the DOS and bandstructures, the broad Ti3d-like bands show a distinct separationinto two sub-bands. This inner conduction band structureis connected to the distorted-octahedron-type short-rangeorder described in the introductory section and descriptivelyderivable from the crystal-field theory of transition-metal(di)oxide complexes [64]. In an octahedral-type crystalfield, the five unoccupied d-states of the central Ti ionare split into the twofold-degenerate eg-like states withdx2−y2 and dz2 character and the energetically lower lyingthreefold-degenerate t2g-like dxy, dyz and dxz type states.

In the O2p-like valence band, a weak two-peak separationis recognizable for rutile and much less prominent for anataseand brookite. The O2p band splitting in rutile results from thesp2-like hybridization in the planar (Y-shaped) OTi3 buildingblocks. The px and py states of the O ion form bondingσ -hybrid states with the central Ti ion. These σ states formthe lower edge of the O2p-like valence band and extendalmost through the whole band. The uppermost valence bandedge consists of pz-type states which form the lone pair(out-of-plane) π states that are higher in energy than thesp2-like hybrid states. The slightly different characteristics

in anatase are due to the more T-like shape of the OTi3building blocks that lead to stronger deviations from the idealsp2 hybrid states than in rutile. In brookite, both rutile-like(Y-shape) and anatase-like (T-shape) OTi3 building blocksare present, which is why the O2p valence band showscharacteristics from both tetragonal phases.

3.2. Optical properties

In figure 4, the dielectric functions including electron–holeinteraction from solution of the BSE are shown for all threemajor phases in ordinary and extraordinary polarization. Theoverall line shape of the dielectric functions is characterizedby an absorption onset above 3 eV and two distinct broadpeaks below and above ∼6.5 eV. This characteristic spectralweight distribution directly reflects the octahedron-typecrystal-field splitting of the Ti3d-like valence states. Thefirst main peak can be identified with interband transitionsfrom the O2p-like valence band into the t2g-like sub-band.The high-energy main peak arises from transitions into theeg-like sub-band. Even for the symmetry-reduced TiO6 unitsof brookite, the formal sixfold coordination leads to identicalqualitative properties.

For rutile, the imaginary part of ε (ω) shows a sharpexcitonic feature for E ⊥ c (cf figure 4(a)) at the adsorptiononset, which is missing in the independent (quasi-)particlespectra on the PBE and G0W0 levels of theory. This first mainpeak is in excellent agreement with the experimental data ofTiwald and Schubert [14]. Also for E ‖ c, the position ofthe first main peak agrees with the experiment. Nevertheless,the peak height is underestimated in our calculation and thepredicted doublet structure of this peak is not resolved inthe experiment. The second major spectral features presentabove 6.5 eV in the calculated spectra for both polarizationsare clearly resolved in the Tiwald–Schubert data. Even thesingle-peak characteristic for E ⊥ c and the double-peakcharacteristic for E ‖ c is seen experimentally. Furtherevidence for notable spectral features above 6.5 eV is alsogiven by electron energy loss spectroscopy measurements ofRocquefelte et al [65].

In light of the newer experimental data and our calculatedspectra, it appears reasonable to argue that the experimentaldata of Cardona and Harbeke [13], commonly used asexperimental reference on TiO2 rutile but not indicating asecond main peak neither for E ⊥ c nor for E ‖ c, do notproperly reflect the absorption characteristics.

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J. Phys.: Condens. Matter 24 (2012) 195503 M Landmann et al

Figure 4. Polarization-dependent real (ε1 (ω)) and imaginary (ε2 (ω)) parts of the complex dielectric function for TiO2 rutile (a), anatase(b) and brookite (c). The dielectric functions have been calculated on the BSE level of theory using 72, 144 and 288 electronic bands, 539,243 and 75 regularly distributed irreducible k-points, and QP energies obtained from G0W0 calculations. The included experimentalreference data were extracted from [14] for rutile and from [15] for anatase.

For TiO2 anatase (cf figure 4(b)) the calculated line shapeof the G0W0–BSE dielectric function is in good agreementwith the experimental data published by Hosaka et al [15] forboth polarizations. The perhaps most characteristic differencefrom rutile is the low-energy shoulder at ∼4 eV and theresulting main peak triplet structure in ε2 (ω) for E ⊥ c. Itshould be noted that this triplet structure exists already in thePBE and G0W0 spectra. For E ‖ c no low-energy shoulderis observable. The lowest energy feature is a very sharpexciton peak at ∼4.5 eV which is slightly blueshifted andslightly overestimated compared to position and intensity inthe experiment. Since the absorption onset for E ‖ c is foundat significantly higher energies, our results indicate a morepronounced optical dichroism for anatase than for rutile.

In contrast to the dielectric tensor of the uniaxialmaterials rutile and anatase, the orthorhombic brookite phaseis a biaxial material with three independent components.Similar to anatase, the absorption edge of brookite is foundto be dichroic due to low-energy features in the two in-planecomponents ε2x (ω) and ε2y (ω) (cf figure 4(c)). In contrastto ε2x (ω), the low-energy feature in ε2y (ω) exists alreadyin the independent-particle spectra at the PBE and G0W0levels of theory, whereas this feature in ε2x (ω) arises fromexcitonic contributions. As observed for anatase, the first mainpeak exhibits a triplet structure. In contrast, the first peakfor E ‖ c is almost structureless, more similar to rutile. Theoverall influence of electron–hole interaction on the spectralweight distribution is less pronounced in brookite than inthe tetragonal phases. This suggests that less local symmetry,which corresponds to smaller spatial anisotropies, diminishesthe formation of sharp exciton features.

Since it is difficult to extract the adsorption onsetfrom the dielectric functions in figure 4, we have

determined the lowest eigenvalues of the electron–holepair Hamiltonian for the three TiO2 materials by arecursive approach [67, 68, 66]. For TiO2 rutile, the lowestG0W0–BSE pair-excitation energy is 3.28 eV. This valueis larger than the fundamental optical bandgap of 3.03 eVdetermined in absorption/photoluminescence spectroscopymeasurements [2–5]. A similar result is obtained for TiO2anatase. While photoluminescence spectroscopy yields afundamental gap of 3.2 eV, the lowest calculated pair-excitation energy in the G0W0–BSE approach is 3.57 eV.Our calculated pair-excitation energies of rutile and anataseagree well with the bandgaps of 3.37 and 3.53 eV thatwere reported by Coronado et al [20] from direct Taucanalysis of UV–visible reflectance spectra. However, thelowest pair-excitation energy of 3.56 eV for brookite in thejust mentioned study is found to be larger than the calculatedlowest pair-excitation energy of 3.13 eV.

4. Concluding remarks

In this study, we have investigated the electronic andoptical properties of the three naturally occurring TiO2polymorphs rutile, anatase and brookite. The more complexorthorhombic brookite phase has, for the first time, beentreated systematically with the same accuracy as the twotetragonal phases. Thus, this work provides TiO2 brookiteHSE06 hybrid functional and G0W0 QP reference data forfurther studies as well as optical spectra from solution of theBSE. Our results indicate that the inclusion of electron–holepair excitations on the BSE level of theory allows an accuratemodeling of the polarization-dependent optical properties.This is especially true for TiO2 rutile, where, in light ofthe more recent data of Tiwald and Schubert [14], the

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J. Phys.: Condens. Matter 24 (2012) 195503 M Landmann et al

reported discrepancies between the calculated and measureddielectric functions seem to be primarily caused by improperexperimental reference data used in the preceding studies.

Acknowledgments

The calculations were done using grants of computer timefrom the Regionales Rechenzentrum of the Universitat zuKoln (RRZK), the Paderborn Center for Parallel Computing(PC2) and the Hochstleistungs-Rechenzentrum Stuttgart.The Deutsche Forschungsgemeinschaft is acknowledged forfinancial support.

References

[1] Dambournet D, Belharouak I and Amine K 2010 Chem.Mater. 22 1173

[2] Pascual J, Camassel J and Mathieu H 1977 Phys. Rev. Lett.39 1490

[3] Pascual J, Camassel J and Mathieu H 1978 Phys. Rev. B18 6842

[4] Amtout A and Leonelli R 1995 Phys. Rev. B 51 6842[5] Tang H, Levy F, Berger H and Schmid P E 1995 Phys. Rev. B

52 7771[6] Mattsson A and Osterlund L 2010 J. Phys. Chem. C 114[7] Bach U, Lupo D, Comte P, Moser J E, Weissortel F, Salbeck J,

Spreitzer H and Gratzel M 1998 Nature 395 583[8] Gratzel M 2005 MRS Bull. 30 23[9] Hagfeldt A, Boschloo G, Sun L, Kloo L and

Pettersson H 2010 Chem. Rev. 110 6595[10] Wyckoff R W G 1963 Crystal Structures vol 1 (New York:

Wiley)[11] Horn M, Schwerdtfeger C F and Meagher E P 1972

Z. Kristallogr. 136 273–81[12] Hermann C, Lohrmann O and Philipp H 1937 Strukturbericht

Band II 1928–1932 (Leipzig: AkademischeVerlagsgesellschaft MBH)

[13] Cardona M and Harbeke G 1965 Phys. Rev. 137 1467[14] Tiwald T E and Schubert M 2000 Proc. SPIE 4103 19[15] Hosaka N, Sekiya T, Satoko C and Kurita S 1997 J. Phys. Soc.

Japan 66 877[16] Sekiya T, Igarashi M, Kurita S, Takekawa S and

Fujisawa M 1998 J. Electron Spectrosc. Relat. Phenom.92 247

[17] Koelsch M, Cassaignon S, Guillemoles J F andJolivet J P 2002 Thin Solid Films 403–404 312

[18] Zallen R and Moret M P 2006 Solid State Commun. 137 154[19] Li J-G, Ishigaki T and Sun X 2007 J. Phys. Chem. C 111 4969[20] Reyes-Coronado D, Rodrıguez-Gattorno G,

Espinosa-Pesqueira M E, Cab C, de Coss R andOskam G 2008 Nanotechnology 19 145605

[21] Wanbiao H, Liping L, Guangshe L, Ghanglin T andLang S 2009 Cryst. Growth Des. 9 3676

[22] Glassford K M and Chelikowsky J R 1992 Phys. Rev. B46 1284

[23] Hardman P J, Raikar G N, Muryn C A, van der Laan G,Wincott P L, Thornton G, Bullett D W andDale P A D M A 1994 Phys. Rev. B 49 7170

[24] Mo S-D and Ching W Y 1995 Phys. Rev. B 51 13023[25] Mikami M, Nakamura S, Kitao O, Arakawa H and

Gonze X 2000 Japan. J. Appl. Phys. 39 L847[26] Asahi R, Taga Y, Mannstadt W and Freeman A J 2000 Phys.

Rev. B 61 7459[27] Calatayud M, Mori-Sanchez P, Beltran A, Pendas A M,

Francisco E, Andres J and Recio J M 2001 Phys. Rev. B64 184113

[28] Muscat J, Swamy V and Harrison N M 2002 Phys. Rev. B65 224112

[29] Hossain F M, Sheppard L, Nowotny J and Murch G E 2008J. Phys. Chem. Solids 69 1820

[30] Wu X, Holbig E and Steinle-Neumann G 2010 J. Phys.:Condens. Matter 22 295501

[31] Shojaee E and Mohammadizadeh M R 2010 J. Phys.:Condens. Matter 22 015401

[32] Zhang Y, Lin W, Li Y, Ding K and Li J 2005 J. Phys. Chem.B 109 19270

[33] Beltran A, Gracia L and Andres J 2006 J. Phys. Chem. B110 23417

[34] Lee B, Lee C, Hwang C S and Han S 2011 Curr. Appl. Phys.11 S293

[35] Janotti A, Varley J B, Rinke P, Umezawa N, Kresse G andVan de Walle C G 2010 Phys. Rev. B 81 085212

[36] Deak P, Aradi B and Frauenheim T 2011 Phys. Rev. B83 155207

[37] Deak P, Aradi B and Frauenheim T 2011 J. Phys. Chem. C115 3443

[38] Lawler H M, Rehr J J, Vila F, Dalosto S D, Shirley E L andLevine Z H 2008 Phys. Rev. B 78 205108

[39] Chiodo L, Garcia-Lastra J M, Iacomino A, Ossicini S, Zhao J,Petek H and Rubio A 2010 Phys. Rev. B 82 045207

[40] Kang W and Hybertsen M S 2010 Phys. Rev. B 82 085203[41] Hohenberg P and Kohn W 1964 Phys. Rev. B 136 864[42] Kohn W and Sham L J 1965 Phys. Rev. A 140 1133[43] Blochl P E 1994 Phys. Rev. B 50 17953[44] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758[45] Kresse G and Furthmuller J 1996 Comput. Mater. Sci. 6 15[46] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett.

77 3865[47] Heyd J, Scuseria G E and Ernzerhof M 2003 J. Chem. Phys.

118 8207[48] Heyd J and Scuseria G E 2004 J. Chem. Phys. 121 1187[49] Heyd J, Peralta J E, Scuseria G E and Martin R L 2005

J. Chem. Phys. 123 174101[50] Heyd J, Scuseria G E and Ernzerhof M 2006 J. Chem. Phys.

124 219906[51] Shishkin M and Kresse G 2006 Phys. Rev. B 74 035101[52] Shishkin M and Kresse G 2007 Phys. Rev. B 75 235102[53] Fuchs F, Furthmuller J, Bechstedt F, Shishkin M and

Kresse G 2007 Phys. Rev. B 76 115109[54] Shishkin M, Marsman M and Kresse G 2007 Phys. Rev. Lett.

99 246403[55] Sham L J and Rice T M 1966 Phys. Rev. 144 708[56] Hanke W and Sham L J 1975 Phys. Rev. B 12 4501[57] Hanke W and Sham L J 1980 Phys. Rev. B 21 4656[58] Albrecht S, Reining L, Del Sole R and Onida G 1998 Phys.

Rev. Lett. 80 4510[59] Rohlfing M and Louie S G 2000 Phys. Rev. B 62 4927[60] Puschnig P and Ambrosch-Draxl C 2002 Phys. Rev. B

66 165105[61] Schmidt W G, Glutsch S, Hahn P H and Bechstedt F 2003

Phys. Rev. B 67 085307[62] Tezuka Y, Shin S, Ishii T, Ejima T, Suzuki S and Sato S 1994

J. Phys. Soc. Japan 63 347[63] Rangan S, Katalinic S, Thorpe R, Bartynski R A,

Rochford J and Galoppini E 2010 J. Phys. Chem. C114 1139

[64] Lucovsky G 2001 J. Optoelectron. Adv. Mater. 3 155[65] Rocquefelte X, Goubin F, Montardi Y, Viadere N,

Demourgues A, Tressaud A, Whangbo M and Jobic S 2005Inorg. Chem. 44 3589

[66] Fuchs F, Rodl C, Schleife A and Bechstedt F 2008 Phys. Rev.B 78 085103

[67] Haydock R 1980 Comput. Phys. Commun. 20 11[68] Benedict L X, Shirley E L and Bohn R B 1998 Phys. Rev. Lett.

80 4514

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