A Unifying Perspective on Oxygen Vacancies in Wide Band Gap

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A Unifying Perspective on Oxygen Vacancies in Wide Band Gap Oxides Christopher Linderälv, Anders Lindman, and Paul Erhart J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b03028 • Publication Date (Web): 21 Dec 2017 Downloaded from http://pubs.acs.org on December 21, 2017

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The Journal of Physical Chemistry Letters

A Unifying Perspective on Oxygen Vacancies in Wide Band Gap Oxides Christopher Linderälv, Anders Lindman, and Paul Erhart∗ Chalmers University of Technology, Department of Physics, Gothenburg, Sweden E-mail: [email protected]

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Abstract Wide band gap oxides are versatile materials with numerous applications in research and technology. Many properties of these materials are intimately related to defects with the most important defect being the oxygen vacancy. Here, using electronic structure calculations, we show that the charge transition level (CTL) and eigenstates associated with oxygen vacancies, which to a large extent determine their electronic properties, are confined to a rather narrow energy range, even while band gap and the electronic structure of the conduction band vary substantially. Vacancies are classified according to their character (deep vs shallow), which shows that the alignment of electronic eigenenergies and CTL can be understood in terms of the transition between cavity-like localized levels in the large band-gap limit and strong coupling between conduction band and vacancy states for small to medium band gaps. We consider both conventional and hybrid functionals and demonstrate that the former yields results in very good agreement with the latter provided that band edge alignment is taken into account. Conduction band minimum

Charge transition level Valence band maximum

Localized

Delocalized

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Wide band gap oxides are versatile materials with applications in many research fields including, e.g., optoelectronic devices, 1,2 catalysis, 3 fuel cells and batteries, 4,5 ion conductors, 6 nonvolatile memories 7 as well as sensors and actuators. 8 Many relevant properties of these materials are intimately related to defects with the oxygen vacancy being arguably the most important one. As a result, an enormous body of work has been directed toward understanding their properties and their impact on materials performance. Since point defects are by their nature very difficult to access experimentally, electronic structure calculations play a crucial role in resolving defect character, thermodynamics, or kinetics. In particular during the last decade, advances in computational techniques and resources along with improvements in the methodology used to analyze defect energetics, 9,10 have led to a very large number of studies pertaining for example to transparent conducting oxides (e.g., ZnO, 9,11–17 In2 O3 , 11,15,18,19 SnO2 15,20,21 ), high-k dielectrics (e.g., ZrO2 , 22 HfO2 23 ), oxide proton and ion conductors, 24 various perovskites (e.g., Refs. 24–27) as well as numerous other oxides (e.g., MgO, 28,29 TiO2 , 30–32 SiO2 , 23 CeO2 33–36 ). This situation begs the question: Are there universal principles that allow us to categorize the properties of oxygen vacancies across many different oxides and possibly anticipate their behavior based on simpler predictors, circumventing the need to carry out explicit calculations for a large number of oxides? A key quantity in this regard, is the charge transition level [CTL, see Fig. 1(a)]. If the CTL is located at least a few tens of an meV below the conduction band minimum (CBM), the vacancy is referred to as deep (as opposed to shallow). Furthermore, its electrons are localized and do not contribute to electrical conduction. Based on an analysis of a few oxides, it has already been shown that the formation of deep as opposed to shallow oxygen vacancies can be connected to the coupling of (metal) cation and oxygen states, with deep character being the result of an antibonding CBM state. 37 Here, we provide a more general perspective. Specifically, we demonstrate that if the electronic structure is properly aligned, certain key features of the oxygen vacancy, most notably the CTL, exhibit a rather small spread across a large number of oxides, representing different lattice structures, stoichiometries, band gaps, and compositions. This finding is at first surprising since,

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as mentioned above, one can expect the electronic character of an oxygen vacancy to be very sensitive to the cation states that compose the conduction band. 37 The observed behavior can, however, be explained by noting that the electronic states associated with the oxygen vacancy are only partially determined by the conduction band states and, especially in very wide gap systems, rather resemble the bound states of a cavity. We also analyze the electronic character of the vacancies (deep vs shallow), which, e.g., in the case of transparent conducting oxides (TCOs), has been a matter of intense scrutiny. 11,15,37–39 To this end, we employ three complimentary criteria that involve formation energies, eigenenergies, and geometry, respectively (Fig. 1). Finally, following up on earlier work, 9,23,40 we demonstrate that the properties of the oxygen vacancy (and in fact many other defects) can be predicted with good accuracy already at the level of conventional exchangecorrelation (XC) functionals (as opposed to hybrid functionals) if band edge alignment as well as the aforementioned indicators are taken into account. We conducted extensive density-functional theory (DFT) calculations using both the vdW-DFcx method, which combines semi-local exchange with non-local correlation, 41,42 and the rangeseparated hybrid HSE06 functional. 43 We note that while the latter yields band gaps that are much closer to experiment, it is also computationally substantially more expensive. We considered 26 different oxides (20 binary and 6 ternary systems; see Fig. 2 and Table S3) representing a broad variety of lattice structures and cations from across the periodic table. Focusing on the neutral (q = 0) and doubly positive (q = +2) charge states, we extracted formation energies, KohnSham (KS) eigenenergy spectra, and geometries from our calculations (Fig. 1). In the subsequent analysis of the results only the oxygen site with the most stable neutral vacancy is included for materials with multiple unique oxygen sites (see Fig. 2). The methodology is described in more detail in the Supplementary Information (SI), which also contains an extended description of the results including the handling of finite-size effects. 10,44,45 We start our analysis with the earth alkali oxides (cations from group 2) that adopt rocksalt structures with the exception of BeO, which prefers tetrahedrally coordinated lattices. These materials represent a wide spread of calculated band gaps from 3.4 eV (BaO) to 9.4 eV (wz-BeO). Yet,

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if the band edges are aligned to the oxygen 1s level 47 (see SI), the vacancy CTL has an average value of −5.66 eV with a standard deviation of only 0.35 eV relative to the vacuum level. (Alignment with the latter has been carried out using data from Ref. 46; for further details see SI). For all neutral vacancies, there is also a localized KS eigenstate in the band gap (see Fig. S3), which has s-character [see Fig. 3(c) for an example]. Extending the analysis to binary oxides in group 3, again one observes the oxygen level to vary rather little between different materials. The average CTL is −5.15 ±0.23 eV. In the group 3 oxides (Sc2 O3 , Y2 O3 , Lu2 O3 ), which all adopt a bixbyite lattice structure, the band gap hardly changes with increasing atomic number of the cation, yet the band edges exhibit some shift. In all three cases, the CTL is inside the band gap and localized KS levels are apparent that as in the case of the group 2 oxides exhibit s-character. Continuing with the group 13 oxides (Al2 O3 , Ga2 O3 , In2 O3 ), one again observes a small variation in the CTL with an average value of −5.37 ± 0.13 eV. This group (together with the Ba-based perovskites) displays the smallest variation in the CTL, which is noteworthy as the band gap varies from 7.9 eV (Al2 O3 ) to 3.7 eV (In2 O3 ). Next, we consider the oxides of group 14, which include SiO2 , GeO2 and SnO2 , where the former two adopt the α-quartz structure. This group display the largest variation, −6.06 ± 0.50 eV, which is mainly due to the transition level in SiO2 being deeper compared to the other materials. The situation for SiO2 (and also for GeO2 ) is a bit more complicated as the oxygen vacancy in this material is subject to extremely large lattice relaxations (see Figs. S4 and S5) for the different charge states, which results in two metastable configurations. 48 The CTL in SnO2 is, however, in close agreement with the total average value. The situation is more complicated in the case of the group 4 oxides (TiO2 , ZrO2 , HfO2 ) but still the CTL variation is modest at −4.39 ± 0.46 eV. Here, ZrO2 and HfO2 have a monoclinic crystal structure and band gaps larger than 5 eV. TiO2 , on the other hand, exhibits several different crystal structures with band gaps closer to 3 eV (here, we included the two most important ones, rutile and anatase). For ZrO2 and HfO2 , the oxygen vacancy CTL is clearly separated from the CBM with a

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localized KS defect state that has predominantly s-character. By contrast, in both polymorphs of TiO2 the CBM is notably lower and is much closer to the oxygen vacancy CTL. Our comparison also comprises six perovskites based on Ba and Pb, respectively. These materials were selected firstly because they are examples of ternary oxides, secondly because they represent different compositions of the bottom of the conduction band (Ti, Zr, and Hf d-states in the Ba series, Pb p-states in the Pb series), and thirdly because as a result of their ferroelectric/antiferroelectric phases VBM and CBM exhibit a notable variation with structure and composition. In the Ba-based perovskites considered here the calculated band gap increases from 3.1 to 5.1 eV when going from the titanate to the hafnate. Yet, as before, the oxygen vacancy CTL exhibits a much smaller variation with an average value of −4.76 ± 0.13 eV. Whereas the CTL coincides with the CBM in the case of BaTiO3 , it is located deep inside the band gap in the case of BaZrO3 and BaHfO3 , mirroring the situation in the binary group 4 oxides. In the Pb-based perovskites the CBM and as a result the band gap exhibits a much smaller variation than in the case of the Ba-based materials, which can be attributed to the contribution from Pb p-states to the conduction band edge. Nonetheless, the oxygen vacancy CTLs are very similar to the Ba-based perovskites with an average value of −4.52 ± 0.29 eV. Finally, we return to the binary oxides and consider the TCOs from groups 12-14: ZnO, In2 O3 , and SnO2 , which have arguably received the most attention. 9,11–21 These materials exhibit some features that require special attention from a computational perspective, such as the position of cation d-band, which is located below the oxygen 2p-dominated valence band. Yet, even in this set of oxides one can observe only a rather small variation of the CTL with an average value of −5.49 ± 0.15 eV. The previous analysis clearly demonstrates that the oxygen vacancy CTL exhibits only small variations among oxides with cations from the same group of the periodic table, provided that the alignment of the band edges is taken into account. Moreover, an alignment of the CTL is even apparent across all oxides included here (Fig. 2), for which one obtains an average value of −5.19 ± 0.66 eV with respect to the vacuum level. To rationalize this behavior and establish the

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conditions that lead to this rather universal behavior, we carefully analyzed not only the energetics but also the electronic and ionic structure of the defects as described in the following. As noted in the introduction, oxygen vacancies, especially in TCOs, have been studied extensively with a key issue being the assignment of deep vs shallow character alluded to above. In this regard, it must be noted that it is important how shortcomings intrinsic to conventional XC functionals are treated. 9 While deep defects are most often identified by a CTL that resides inside the band gap, this indicator alone can be misleading if the separation between band edge and CTL is within the error bar of the calculation, an all too common occurrence. As discussed below a deep CTL can also be hidden due to an underestimation of the band gap. To resolve these issues, we consider three complimentary indicators that in combination enable a clean identification of deep and shallow vacancies. In addition to (1) a CTL in the band gap [Fig. 1(a)], these are (2) the presence of KS eigenlevels inside the band gap in at least one charge state [usually the neutral one; see Fig. 1(c)] and (3) a large difference in relaxation behavior between different charge states [Fig. 1(b)]. We find that in particular the last measure serves as a very reliable indicator that furthermore transcends limitations of conventional XC functionals in correctly describing band gap and band edge energies. Based on the CTLs and KS eigenlevels from our conventional XC functional calculations (Figs. S1 and S3), most of the oxygen vacancies considered here should be considered deep defects. However, for BaO, the TiO2 polymorphs, the perovskites with the exception of BaHfO3 , as well as SnO2 and In2 O3 , the distinction is more difficult. For these materials, hybrid XC functional calculations indicate that the perovskites except for the titanates as well as In2 O3 are deep while for the remaining cases a shallow character appears more likely (Fig. 2). The geometric relaxation of the different charge states, however, provides by far the clearest classification. By way of introduction let us consider the following: In the case of the doubly charged oxygen vacancy, the eigenstates are occupied up to the VBM. Locally, an O2− ion has been removed and in response the cations surrounding the vacancy typically relax slightly outward (away from the vacant site). In the case of the neutral vacancy, we can distinguish two types

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of behavior. If the vacancy is shallow, the two electrons associated with the defect are donated to the conduction band and delocalize over the entire system. As a result, the electronic charge distribution as well as the ionic positions around the vacant site barely change compared to the charged vacancy structure [red line Fig. 1(b)]. By contrast, in the case of a deep vacancy, the two excess electrons are localized and affect the local charge distribution. In response, the surrounding cations relax toward the vacancy site, effectively “bonding” to the localized electrons [blue line in Fig. 1(b)]. This relaxation mechanism enables a simple and, more importantly, clear assignment of vacancy character. As a quantitative measure, we can use for example the largest relaxation of any atom in the system compared to the ideal lattice (Fig. S4) or the maximum change of any ionic position between the two charge states (Fig. S5), both of which result in the same classification (colored circles in Fig. 2). All of the cases that are identified above as deep according to CTL or KS defect level position exhibit a pronounced difference with respect to the relaxation of the neighborhood of the vacancy in the doubly charged and neutral charge states. In addition, a marked difference is apparent for SnO2 suggesting a deep defect, whereas for the TiO2 polymorphs as well as the titanates, the different charge states show virtually identical relaxation patterns, indicative of shallow defects. We note that while electron polarons and their interaction with oxygen vacancies have been reported for the rutile phase of TiO2 , 30,31 they do not lead to a deep cavity-like vacancy state. A detailed discussion of this aspect can be found in the SI (Sect. S-II D). It must be stressed that the relaxation behavior of the vacancies discussed here is already captured using conventional XC functionals, which implies that the classification of deep vs shallow character should be possible without resorting to computationally much more demanding hybrid XC calculations. It has been previously observed that there is a simple correlation between defect formation energies from conventional and hybrid XC functionals 23 and that localized defect levels are usually only weakly affected by band edge shifts either due to the addition of local (via DFT+U ) or exact exchange. 9,23 As will be discussed now, the present results provide further support for this notion and clarify the behavior of conventional vs hybrid functionals.

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In the case of a deep oxygen vacancy such as the one in MgO, the formation energies from conventional and hybrid XC functional calculations are practically identical for both the doubly charged and the neutral charge state, provided that the offset between the VBM from conventional and hybrid XC functionals is included [Fig. 3(a)]. In the case of a shallow vacancy such as the one in BaTiO3 , however, reasonable agreement is only obtained for the charged defect. By contrast, the neutral vacancy formation energy from the conventional functional (vdW-DF-cx) is underestimated by approximately two times the CBM shift while the CTL tracks the conduction band edge [Fig. 3(b)]. The underestimation is thus clearly related to the underestimation of the CBM position (and implicitly the band gap). Conventional functionals are well known to have issues with the description of band gaps and the absolute position of band edges. 49 In many cases, they succeed, however, in describing localized states that are either fully empty or fully occupied. 11,50,51 This can also be observed here, as the energetic position of the localized defect states associated with deep vacancies is hardly affected by the addition of exact exchange. 23 These effects carry over to the formation energies. Most notably, the formation energies for the neutral charge state of deep vacancies are in close agreement between conventional and hybrid functionals [open symbols in Fig. 4(a)]. In the case of shallow vacancies, however, one observes larger deviations that follow the shift of the conduction band edge [filled symbols, also see Fig. 3(b)]. In the case of the doubly charged vacancies, the conventional functional calculations systematically overestimate the formation energies [blue circles in Fig. 4(b)]. If one accounts, however, for the shift of the VBM one again obtains close agreement, with shallow and deep vacancies showing a similar level of accuracy [orange diamonds in Fig. 4(b)]. It should be recalled that also hybrid functionals can be at times unreliable when it comes to the prediction of band edge shifts. 49 Yet, the separation of the error in terms of band edge shifts in conjunction with an understanding of deep vs shallow character and more reliable methods for calculating band edge positions using, e.g., G0 W0 calculations enables one to assess these errors

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in a controlled manner. 9,17,45,52 We furthermore would like to note that there are situations when conventional functionals fail qualitatively, e.g., when partially occupied states are involved, such as for polarons. 47,51,53,54 In such cases, alternative approaches must be employed to overcome the intrinsic qualitative shortcomings, which can but need not involve exact exchange. 51,53 We are now in a position to address the physical mechanism behind the alignment of the oxygen vacancy CTL across many oxides, the variation of which is much smaller than that of band gaps and CBM positions (Fig. 2). It has been demonstrated previously for some oxides that the oxygen vacancy states are derived from conduction band states and are thus associated with the cations that surround the vacant site. 37 This picture suggests a rather strong correlation between CBM and vacancy level, which is not observed here. It is now instructive to consider the earth alkaline oxides as they both have very simple lattice structures and a large variation in band gap. For example, in the case of MgO the vacancy level exhibits a very clear s-character with negligible contributions from the surrounding cations [Fig. 3(c)]. This indicates that the defect state should rather be described as the bound state of a three-dimensional cavity1 . The effective size of an oxygen ion is primarily determined by its charge state. More quantitatively, the difference in local potential between vacancy and ideal configuration exhibits only minor variations between different oxides (see Fig. S7). The extent and depth of the vacancy potential are thus only weakly dependent on the oxide, which translates to the energetic position of the first bound states. In this context, it is also noteworthy that a recent machine learning study that considered the formation energies of neutral vacancies arrived at a model that include neither the atomic sizes nor the coordination as descriptors although both of them were included during training. 58 The above considerations suggest that in the limit of a very high conduction band edge, the lowest electronic state associated with the vacancy, from which other properties are derived, should become independent of the cation. As the conduction band moves downward, cation derived conduction band levels will start to hybridize with the vacancy, which ultimately leads to the emergence of shallow vacancy states. The present calculations suggest that this condition is obtained 1

This is reminiscent of the Mollwo-Ivey relationship, which postulates that the absorption energy Eabs of anion vacancies in the alkali halides scales with the lattice constant a as Eabs ∝ a−n , where n is close to two. 55–57

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if the CBM drops to about −5 to −4 eV relative to the vacuum level. At this point, cation-oxygen coupling becomes important and can give rise to more subtle yet important contributions, 37 which play a role for example in the case of the titania polymorphs as well as the titanates. The behavior of the CTLs in oxides should be compared to the universal alignment observed for the (+1/ − 1) CTL of hydrogen interstitials in semiconductors. 46 The latter was found to be located −4.5 eV below the vacuum level (about 0.7 eV higher in energy compared to the average oxygen vacancy CTL) with a very small variation about the mean value. This behavior can be understood by considering the small size of the hydrogen atom, which effectively acts as a smaller perturbation on the electron density (and lattice) than an oxygen vacancy. Because of the relatively small perturbation, the insertion of a hydrogen atom in an interstitial region of a crystal allows for a more accurate probing of the local potential. By comparison, in the present case the spread of CTL values is markedly larger. This reflects the fact that the vacancy “cavity” is larger than a hydrogen atom and thus is more sensitive to variations in electron density and potential in its surrounding. Nonetheless, the comparison with the hydrogen interstitial case provides a perspective on and further rationalization for the present case. In this work we have discussed three indicators to distinguish between shallow and deep vacancy character. Specifically, we considered (1) CTL, (2) geometric relaxation, and (3) KS levels. Our analysis has shown that when applied with care these measure yield consistent results. Among the three indicators considered the geometric relaxation serves as the most reliable predictor, which already works well at the level of conventional XC functional calculations. On the other hand, the position of the CTL relative to the CB edge is much more sensitive to the description of the electronic structure of the ideal reference material and thus the parametrization of the computational method employed (e.g., mixing parameters in the case of hybrid functionals or U parameters in the case of DFT+U ). This reflects the well-known fact that the correct description of the absolute and relative values of electron affinities and ionization potentials for different materials can require much more refined techniques such as the GW method. While we can thus conclude that at the level of hybrid XC functionals the description of the CBM-CTL separation is sensitive to

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computational details, 59 the categorization of the vacancy characters is unaffected by these issues. We note that the insights achieved in the present work can also provide guidance for identifying good descriptors for machine learning vacancy properties. 58,60 Here, the ultimate objective is to overcome the need to conduct fully fledged quantum mechanical calculations that are tied to specific constituents, compositions and structures, and to build models that can extrapolate for example to both crystalline and non-crystalline mixed oxide systems with several cation species. The standard deviation of the CTL averaged over all oxides in this work exhibits a standard deviation of 0.66 eV, which is not much larger than the predictive error of 0.39 eV reported for a machine learning model for neutral vacancies. Knowledge of CTL, neutral vacancy formation energy, and absolute VBM position are sufficient to compute the formation energy of the charge vacancy. The existing data is thus already sufficient to obtain a simple prediction of this important material property. In conclusion, in the present work we have conducted a systematic exploration of the properties of oxygen vacancies in binary and a few ternary oxides. It was shown that the vacancy CTL in these wide gap semiconductors is confined to a relatively small range, especially to the variation in band gap, cation species, and lattice structure. This relative alignment effect can be attributed to the fact that the effective size of the oxygen anion is very similar for all compounds considered here, and, as a result, also the vacancy “cavity”. In the very large gap limit, the cavity effect overrides the coupling to the conduction states, which, however, becomes dominant as the CBM position approaches a value of −5 to −4 eV relative to the vacuum level. In analyzing the vacancy character, we established the usefulness of combining CTL, eigenenergy spectrum, and defect geometry for distinguishing deep vs shallow character. Furthermore, complementing earlier work, we demonstrated that reliable formation energies can already be obtained using conventional XC functionals, provided that defect character and band alignment are properly taken into account. We expect that the trends and insights achieved in the present work will provide a very good starting point for identifying good descriptors for machine learning of CTLs or formation ener-

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gies of charged vacancies. The present work thus provides the basis for a much more systematic approach to predicting and tuning the properties of vacancies in particular and defects in general.

Acknowledgement We would like to acknowledge the Knut and Alice Wallenberg Foundation for financial support. Computational resources have been provided by the Swedish National Infrastructure for Computing (SNIC) at PDC (Stockholm) and NSC (Linköping).

Supporting Information Available The following files are available free of charge. • Supplementary_Information.pdf: Details concerning the applied methodology for densityfunctional theory calculations and an extended description of the results. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Formation energy

sha

shallow deep

de

ep

CTL

c

deep

Quasi-particle energy

b shallow charged

CB

llow

Formation energy

a VB

neutral

Electron chemical potential

Configuration coordinate

Figure 1: Defect characteristics. Schematic illustrating the connection between formation energies, charge transition levels (CTL), electronic eigenenergy spectra, and defect geometries for deep and shallow defects.

Group 2

Group 3

Group 4

Ba-perovskites

Pb-perovskites Group 12

Group 13

Group 14

2 0 −2 −4 −6 −8 −10

Figure 2: Charge transition levels (2+/0) of oxygen vacancies. For materials with multiple unique oxygen sites (ZrO2 , HfO2 , PbZrO3 , PbHfO3 , Ga2 O3 ), black lines correspond to the CTL for the oxygen site at which the neutral vacancy is most stable while green lines mark the remaining ones. Calculations were performed using a hybrid XC functional at the equilibrium structure obtained from a conventional XC functional. The positioning of the valence and conduction band edges reflects the alignment of different oxides with respect to the O 1s-level. The origin of the energy scale has been set to the vacuum level by alignment of the SiO2 VBM with the corresponding VBM in Fig. 2 of Ref. 46. The dashed line and the shaded grey area indicate the average CTL with one standard deviation. The coloring of the circles indicates deep (blue) vs shallow (red) character according to the difference in ionic positions between the two charge states (see Supplementary Information for details).

20


2

SiO 2 Ge O 2 Sn O

2O 3

2O 3

In

2O 3

Al

Ga

O Zn

3

TiO Pb 3 ZrO Pb 3 HfO

Pb

3

Ba TiO Ba 3 ZrO Ba 3 HfO

2

2

HfO

ut)

na ZrO )

2 (a

TiO

2 (r

TiO

2O 3

Lu

3

O

2O 3

Y2

O Ba

Ca

Sc

Be

Be O

O SrO

−12

(w z) O( zb ) Mg O

Energy wrt vacuum level (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Formation energy (eV)

8

a

MgO

6

6

conventional

b

BaTiO3

4

hybrid

4

2 2

conventional hybrid

0 0

0

2

4

6

0

1

2

3

Electron chemical potential (eV)

c

d

Figure 3: Deep vs shallow vacancy character. (a,b) Oxygen vacancy formation energy as a function of electron chemical potential calculated with conventional and hybrid functionals for (a) MgO and (b) BaTiO3 . (c,d) Charge density of the lowest defect state for the neutral vacancy in (c) MgO and (d) BaTiO3 . In the case of MgO the vacancy state is located 2.48 eV below the CBM and occupied, whereas in the case of BaTiO3 it resides 0.93 eV above the CBM and is unoccupied. The isovalues of the charge densities are set to 0.01 and 0.05 in both cases.

Hybrid formation energy (eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8

a neutral

3

7

2

6

1

5

0

4

−1

3

−2 3

4

5

6

7

8

b charged as-calculated shifted

−2 −1

0

1

2

3

Formation energy from conventional XC functional (eV)

Figure 4: Oxygen vacancy formation energies. Comparison of formation energies for (a) neutral and (b) doubly charged oxygen vacancies calculated using a conventional and a hybrid XC functional. Open and filled symbols indicate data for deep and shallow vacancies, respectively, according to the classification shown in Fig. 2.

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A Unifying Perspective on Oxygen Vacancies in Wide Band Gap

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