Computational Materials Science 118 (2016) 309–315

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Oxygen vacancy diffusion in bulk SrTiO3 from density functional theorycalculations

http://dx.doi.org/10.1016/j.commatsci.2016.02.0410927-0256/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: xhx@utk.edu (H. Xu).

Lipeng Zhang a, Bin Liu a, Houlong Zhuang b, P.R.C. Kent b,c, Valentino R. Cooper d, P. Ganesh b,Haixuan Xu a,⇑aDepartment of Material Science and Engineering, The University of Tennessee, Knoxville, TN 37996, USAbCenter for Nanophase Materials Science, Oak Ridge National Laboratory, Bethel Valley Road, Oak Ridge, TN 37831, USAcComputer Science and Mathematics Division, Oak Ridge National Laboratory, Bethel Valley Road, Oak Ridge, TN 37831, USAdMaterials Science and Technology Division, Oak Ridge National Laboratory, Bethel Valley Road, Oak Ridge, TN 37831, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 January 2016Received in revised form 24 February 2016Accepted 28 February 2016Available online 1 April 2016

Keywords:DFT methodPerovskite structureOxygen vacancyDiffusion energy barrierBoundary condition

Point defects and their diffusion contribute significantly to the properties of perovskite materials.However, even for the prototypical case of oxygen vacancies in SrTiO3 (STO), the predictions of oxygenvacancy activity vary widely. Here we present a comprehensive and systematic study of the diffusion bar-riers in bulk STO. Using density functional theory (DFT), we assess the role of different supercell sizes,density functionals, and charge states. Our results show that vacancy-induced octahedral rotations,which are limited by the boundary conditions of the supercell, can significantly affect the computed oxy-gen vacancy diffusion energy barrier. In addition, we find that the diffusion energy barrier of a chargedoxygen vacancy is lower than that of a neutral one. This difference is magnified in small supercells.We demonstrate that with increasing supercell size, the effects of the oxygen vacancy charge state andthe type of DFT exchange and correlation functional diminish, and all DFT predicted migration energybarriers asymptote to a range of 0.39–0.49 eV, which is smaller than the reported experimental values.This work provides important insights and guidance that should be considered for investigations of pointdefect diffusion in perovskite materials and in oxide superlattices.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Ceramics with a perovskite-type structure (general formulaABO3) consist of a network of corner-sharing BO6 octahedra, whichform higher-coordination sites for the A-cations in a roughly cubicarray. The wide variety of chemical compositions and atomicdefects that can be accommodated in this structure give rise tothe rich variety of phenomena observed in perovskite oxides. Asa prototypical perovskite, SrTiO3 (STO) attracts considerable atten-tion due to its good insulating properties, excellent optical trans-parency, and high dielectric constants [1,2]. It has also beendemonstrated that STO shows potential capacitive [3] and resistiveswitching properties [4–7]. In addition, STO has been extensivelyused as a substrate for superlattices and is the parent oxide thatleads to two-dimensional electron gases (2DEGs) [8,9] in manyoxide heterointerfaces.

Oxygen vacancies (VO) are representative intrinsic defects that,in many cases, play an important role in the physical properties of

STO. An VO is generally produced during crystal growth or filmdeposition processes with nonstoichiometric reactants, as a resultof oxygen gas release accompanying heat treatment, and in dopedSTO serving as a charge compensation mechanism. They formreadily within STO and have been indicated to be the driving forcebehind STO-associated blue and green light emission [10,11] andhigh n-type conductivity [12–15]. Therefore, fully understandingthe stability and diffusion of VO is critical for tailoring the physicalproperties of STO for future applications. However, while the ener-getics and stability of oxygen vacancies and their effects on theproperties of STO are understood, less is known about the energet-ics of VO diffusion.

Numerous efforts have sought to study the diffusion of VO inSTO, producing values of VO migration energy barriers (MEBs) thatvary significantly. For instance, the activation energy of VO diffu-sion was estimated to be 0.60–0.98 eV based on different experi-mental measurements [16–18]. Even different density functionaltheory (DFT) calculations exhibit significant deviations betweenthe calculated values of the oxygen vacancy diffusion barrier. TheDFT estimations of the energy barrier range from 0.35 to 0.9 eV[19–26]. The deviations in DFT results are considerable and the

310 L. Zhang et al. / Computational Materials Science 118 (2016) 309–315

underlying cause for such deviations is unclear, as different simu-lation setups and parameters have been employed in the previouscalculations. Carrasco et al. [22] calculated neutral VO migrationenergy barriers in different STO supercells containing 40–180 atoms with different shapes, predicting a range of values from0.35 to 0.7 eV, depending on the size and shape of the supercell.Evarestov et al. [26] performed similar work using the VASP code.They found that the migration energy barrier mainly decreaseswith increasing supercell size. However due to computational lim-itations, the largest STO supercell they studied contained270 atom. Al-Hamadany et al. [23] explored charged and neutraloxygen vacancy migration in STO using spin-polarized DFT withthe local density approximation (LDA). They employed STO super-cells containing up to 320 atoms. The energy barriers in the largestcells, containing 320 atoms, were 0.81 and 0.77 eV for the neutraland charged (2+) oxygen vacancy, respectively. Lontsi-Fomenaet al. [24] investigated oxygen vacancy diffusion in supercells of40 atoms using the Perdew–Burke–Ernzerhof (PBE) functional,finding the migration energy barrier for a charged (2+) oxygenvacancy to be 0.58 eV. In order to consider the localization of delectrons, Cuong et al. [25] employed an LDA + U approach withU–J = 2.3 eV and STO supercells of 135 atoms, finding a barrier of0.6 eV. These large differences emphasize the fact that calculatedVO diffusion energy barriers depend on the chosen DFT functional,supercell size, and other possibly unknown factors. As such, a com-prehensive understanding of the different factors contributing tothese deviations is required.

In this work, VO formation energies and migration energy barri-ers in STO were calculated in a series of STO supercells with differ-ent number of unit cells, DFT exchange–correlation functionals,and vacancy charge states to systematically examine their effectson VO diffusion. We find that the supercell size, VO charge stateand Hubbard-U value all affect the predicted migration energy bar-riers of isolated oxygen vacancies in small STO (2 � 2 � 2) super-cells. However, with increasing supercell size, the effect of chargestate and the type of exchange–correlation functionals diminishsignificantly, suggesting that the deviations are dominated byfinite size effects. The calculated oxygen vacancy diffusion energybarriers with different DFT methods tend toward a stable numberwhen the STO supercell is sufficiently large. Given that addition ofa Hubbard-U term is thought to give a more accurate representa-tion of the electronic states of oxygen vacancies within the STOsupercell, it stands to reason that PBE + U and PBEsol + U will pre-dict more reliable oxygen vacancy diffusion barriers. Furthermore,our calculations demonstrate that the suppression of octahedralrotations brought about by artificial boundary conditions canseverely alter predicted diffusion barriers. Our work provides use-ful guidelines for simulations of the energetic barriers of defectmigration in complex oxides with electronic structure and atomis-tic methods.

2. Methodology

The Vienna Ab-initio Simulation Package [27] (VASP) code usingthe projector augmented wave (PAW) method is used in the pre-sent density functional theory (DFT) calculations [28]. The corre-sponding {core}/valence configurations for each element are Sr:{[Ar]3d}/4s4p5s; Ti: {[Ne]3s}/3p3d4s; O: {[He]}/2s2p. The cutoffenergy is chosen to be 550 eV based on convergence tests of0.01 eV/atom. The Perdew–Burke–Ernzerhof (PBE) [29] form ofthe generalized gradient approximation (GGA) and the revised ver-sion for solids, PBEsol [30], are chosen as the exchange–correlationfunctionals. In addition, the rotationally invariant Hubbard-U termintroduced by Liechtensitein et al. [31] was also applied. We stud-ied an isolated oxygen vacancy in 2 � 2 � 2, 3 � 3 � 3, 4 � 4 � 4,

and 5 � 5 � 5 STO supercells with a cubic Pm�3m perovskite struc-ture and used 2 � 2 � 2 K-points following the Monkhorst–Packscheme [32]. The K-point mesh was converged to give an energydifference of less than 0.01 eV between the 2 � 2 � 2 and4 � 4 � 4 K-point meshes. The structural relaxations were per-formed until all the Hellmann–Feynman forces are smaller than0.01 eV/Å. To correct for the on-site Coulomb interaction of the3d orbitals, the rotationally invariant + U method [30] is appliedwith U = 5.0 eV and J = 0.64 eV [33] to both exchange–correlationfunctionals. Although different values could be tested, these valuespredicted a larger band gap, which was closer to the experimentalvalue of 3.2 eV for bulk STO [34]. The VO position and diffusion pathare depicted in Fig. 1. Because of the symmetry of the cubic struc-ture, there is just one unique diffusion path (Fig. 1(b) depicts thediffusion path in a representative 3 � 3 � 3 supercell). The diffu-sion paths for neutral and charged vacancy in these supercellsare indistinguishable.

The oxygen vacancy formation energy is defined as:

Ef ðVqOÞ ¼ EtotðVq

OÞ � EtotðBulkÞ �X

i

nili þ qðEF þ Ev þ DVÞ ð1Þ

where Ef ðVqOÞ is the total energy derived from a supercell calculation

with one oxygen vacancy in charge state: q, Etot (Bulk) is the totalenergy for the reference perfect bulk supercell; ni indicates thenumber of atoms of type i that have been added to (ni > 0) orremoved from (ni < 0) the supercell when the defect is formed;and li is the corresponding chemical potential of each species. EFis the Fermi level, referenced to the valence-band maximum Ev inthe bulk. DV is a correction term, to align the reference potentialin our defect supercell with that in the bulk [35]. In this work, theformation energy of a neutral V0

O and charged V2þO were calculated.

The chemical potential of oxygen is chosen to be ½ EO2 (equilibriumgas state), when the PBEsol + U functional were applied, the value ofDVs in 2 � 2 � 2, 3 � 3 � 3, 4 � 4 � 4 and 5 � 5 � 5 are calculatedas 0.16 eV, 0.06 eV, 0.02 eV and 0.03 eV, respectively.

The migration barriers of the neutral and charged oxygenvacancy were both computed by the climbing-image nudged elas-tic band (CI-NEB) method [36]. We first tested up to 5 images in a2 � 2 � 2 STO supercell and found that the calculated migrationenergy barrier is not sensitive to the number of images, which isattributed to the simple chemical environment along the diffusionpath and success of the climbing image algorithm. For the3 � 3 � 3 and 4 � 4 � 4 supercells we used 3 images, and 1 imagefor the most computationally costly 5 � 5 � 5 supercell.

3. Results and discussion

3.1. Lattice constant

Prior to the defect study, the geometry of STO was optimizedusing different exchange–correlation functionals. In all calcula-tions, no distortions were found in the perfect optimized STOsupercells. The calculated lattice constants are summarized inTable 1. These values are comparable with other simulations[37,34] and experimental results (3.905 Å [38]). Among these func-tionals, PBEsol [30] predicts a lattice constant, which is the closestto the experimental value. PBEsol + U, PBE, and PBE + U functionalsoverestimate the lattice parameter (<1%). These lattice constantsare used for subsequent calculations of the energetics of oxygenvacancy formation and diffusion barriers.

3.2. Defect formation energy (DFE)

It is known that the VO formation energy depends on the oxygenchemical potential and electronic Fermi level [39]. Here, the oxy-

Fig. 1. (a) Isolated VO in a 3 � 3 � 3 STO supercell; (b) the oxygen vacancy diffuses to the adjacent oxygen position; (c) VO diffusion path in a 3 � 3 � 3 STO supercell. Thegreen, red, and blue balls represent Sr, O, and Ti atoms, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

Table 1Computational and experimental values of STO unit cell lattice constant.

DFT functional PBEsol PBEsol + U PBE PBE + U PBE PW B3LYP LDA Exp.

Lattice constant (a/Å) 3.898 3.939 3.940 3.969 3.940 [36] 3.950 [33] 3.940 [33] 3.860 [33] 3.905 [37]

Fig. 2. Relative formation energy of an oxygen vacancy as a function of STOsupercell size. The oxygen vacancy formation energy in the 2 � 2 � 2 supercell isused as reference and is set to 0. Bold and dash lines denote data of a neutral andcharged vacancy, respectively.

L. Zhang et al. / Computational Materials Science 118 (2016) 309–315 311

gen chemical potential was chosen as �4.338 eV, which is half ofthe chemical potential of the O2 molecule. We applied the methodof reference [39] to calculate the Fermi level energy, assuming theFermi level is at the conduction band minimum. In addition, theinteraction from periodic images was estimated based on the for-mulism in [40] and this interaction is found to be rather small(<1.2% in the 2 � 2 � 2 STO supercell). Therefore, this correctionterm [40] was not included in our calculations of formation ener-gies. The oxygen vacancy formation energies of neutral andcharged (2+) states in different supercells are listed in Table 2.The relative formation energies versus supercell size obtained withthe PBEsol + U functional are plotted in Fig. 2. The oxygen vacancyformation energy in a 2 � 2 � 2 supercell is used as a reference andis set to 0. It is observed that the DFEs decrease with increasingsystem size; when the supercell is larger than 4 � 4 � 4, the DFEsis approaching to convergence, which is consistent with otherstudies on defect stabilities [35]. The DFEs drop by �0.3 eV(PBEsol + U) for the neutral oxygen vacancy when the supercell isenlarged from 2 � 2 � 2 to 4 � 4 � 4. For the charged oxygenvacancy, DFEs decrease by �1.6 eV using PBEsol + U. Thesedecreases in DFEs are attributed to the long-ranged relaxation ofatoms caused by VOs in the supercell. The effect of electron local-ization in the VO site may be more accurately described using thePBEsol + U, because the charge density is strongly localized onthe two Ti atoms adjacent to the oxygen vacancy. Buban et al.[35] found that the DFEs of neutral and charged (1+, 2+) VOsdropped by roughly 1.1–1.38 eV when going from a 40- to a320-atom supercell, which is comparable with our value of1.6 eV calculated with the PBEsol + U functional on a charged oxy-gen vacancy. The results of PBE with spin polarization, PBEsol, andPBE + U show similar trends as described above.

Table 2Oxygen vacancy formation energy (eV) in different supercells.

Charge state 2 � 2 � 2 3 � 3 � 3 4 � 4 � 4 5 � 5 � 5

VO (0) 6.491 6.576 6.177 6.095VO (2+) 8.296 6.943 6.634 6.219

3.3. Migration energy barrier

3.3.1. Effect of supercell size on oxygen vacancy migration energybarrier

Fig. 3 shows the migration energy barriers of a neutral vacancyobtained using different supercell sizes. Similar to DFEs, the MEBsdecrease as the supercell size increases. The MEB determined usinga 2 � 2 � 2 supercell deviates considerably from those of largersupercells, indicating that this supercell size is too small to predictthe MEB accurately. When the system size reaches to 5 � 5 � 5, thespread of DFT values is significantly reduced. For a single oxygenvacancy in STO, the relaxations, which appear as a long-rangedinward relaxation, extend far from the oxygen vacancy site(�16 Å) [35]. As the boundaries of the larger STO supercells areapproached, the relaxation of these atoms far from the vacancy isvery small. When the STO supercell size increases to 5 � 5 � 5,

Fig. 3. Oxygen vacancy diffusion energy barriers as a function of supercell size. Thedashed and bold lines stand for the results calculated with +U and without +U,respectively; black denotes the calculated values using the PBE functional and reddenotes the calculated values using the PBEsol functional.

312 L. Zhang et al. / Computational Materials Science 118 (2016) 309–315

the MEBs of an oxygen vacancy approaches convergence, as Fig. 3illustrates. The range of converged values, 0.39–0.49 eV, underesti-mates the MEB of an isolated oxygen vacancy derived from exper-iments and some other computations of �0.60 eV [18,24,25]. Theunderestimation relative to other simulations primarily comesfrom STO supercell size effects. Previously, it was shown thatincreased interactions between oxygen vacancies result in anenhancement in the oxygen migration energy barriers [25]. Inour largest supercell (5 � 5 � 5, containing 625 atoms), the inter-action between two adjacent oxygen vacancies is significantlylower than in the smaller supercells (up to 320 atoms) previouslystudied [23–25]. As such, the migration energy barriers for the5 � 5 � 5 supercell are expected to be lower than earlier simula-tions which employed much smaller supercells. The reason forthe underestimate when comparing with the experimental result

(a)

(d) (c)

SupercellBond ang

IS 2 2 2 86.463 3 3 88.244 4 4 87.425 5 5 88.08

A

B

C 2.94 Å

3.07Å

Ti1

Ti2

A

B

C

Ti3

Ti4

B Ti1

Ti2

Ti3

Ti4

Fig. 4. (a) Oxygen octahedral rotations induced by an oxygen vacancy in a 4 � 4 � 4distance between the diffusing oxygen and its neighboring atoms in the STO supercell.states in the (c) 3 � 3 � 3 and (e) 4 � 4 � 4 supercells and in its transition states in the (d)oxygen atoms, respectively. D(OA–OB) and D(OA–Ti2) represent the distance between a d(For interpretation of the references to color in this figure legend, the reader is referred

of �0.6 eV may be due to limitations of the standard DFT methodwe employed in our calculations or the interactions of a single oxy-gen vacancy with complex defects in the experimental sample.Hybrid functionals such as the B3PW xc-functional have beendemonstrated to predict a more accurate optical gap (3.65 eV) forbulk STO [41]. However, it is still a huge challenge of computa-tional cost to use such approaches to predict the vacancy migrationenergy barrier in very large STO supercells in which the vacancyconcentration can reach a value comparable to experiment.

For PBEsol or PBEsol + U, the energy barriers calculated in3 � 3 � 3 STO supercells are lower than that of the 4 � 4 � 4 super-cell. We found that this comes from the difference of the distortionmodel of the local atomistic environment around the oxygen diffu-sion path in 3 � 3 � 3 and 4 � 4 � 4 supercells. Due to the absenceof one oxygen atom in STO bulk, oxygen octahedral rotations areintroduced into the system [11]. This local configuration changeis limited by the boundary condition of the supercell, leading tothe two different situations observed in the 3 � 3 � 3 and4 � 4 � 4 supercells. One situation makes the distances betweenthe diffusing oxygen atom and its nearest oxygen neighbor longer,while the other shortens it, resulting in substantially differentMEBs.

The initial and transition point configurations of the 3 � 3 � 3and 4 � 4 � 4 STO supercells are shown in Fig. 4. An oxygen octa-hedral rotation (a�b�c0) pattern (following Glazer pattern [42]) isfound when an oxygen vacancy is introduced into the system(shown as Fig. 4(a), in a 4 � 4 � 4 supercell). Here, we use the bondangle of OB–Ti1–OC, which is near the oxygen vacancy (shown asFig. 4(c) and (e)) to characterize the oxygen octahedral rotationmagnitude in different sizes of STO supercells. The magnitudes ofthis angle in the initial state (IS) and transition state (TS) are shownin Fig. 4(b), where it is evident that boundary conditions controlthe rotational characteristics. For the initial states, when the num-ber of unit cells in a particular direction is even, bond angles of OB–Ti1–OC are smaller with values of 86.46�, 87.42� in 2 � 2 � 2 and4 � 4 � 4, respectively. On the other hand, for an odd number ofunit cells, these bond angles are larger with values of 88.24�,88.08� in 3 � 3 � 3 and 5 � 5 � 5, respectively. Similar observa-

A B

C

Ti1

Ti2

Ti3

Ti4 3.07 Å

A

B

C

Ti1

Ti2

(b)

le(OB-Ti1-OC D(OA-OB)/Å D(OA-Ti2)/ÅTS 88.15 2.84 3.07 95.76 2.94 3.07 93.43 2.89 3.07 96.47 2.95 3.09

(e) (f)

STO supercell; (b) oxygen octahedral rotational characteristic parameters and theThe oxygen vacancy diffusion plane (010) showing an oxygen vacancy in its initial3 � 3 � 3, (f) 4 � 4 � 4 STO supercells. The blue and red balls represent titanium andiffusing oxygen atom A and an oxygen atom B or titanium atom 2 in Fig. 4(d) or (f).to the web version of this article.)

Fig. 5. Diffusion energy barrier of neutral and charged oxygen vacancies fordifferent STO supercell sizes calculated with PBEsol functional.

L. Zhang et al. / Computational Materials Science 118 (2016) 309–315 313

tions are found for the transition states, where the OB–Ti1–OC bondangles are again smaller with values of 88.15�, 93.43� in even-numbered supercells (2, 4, respectively) and larger, i.e., 95.76�,96.47�, in odd-numbered supercells (3, 5, respectively). In the2 � 2 � 2 supercell, this angle is much lower than those in othersupercells, because the corresponding oxygen rotations are pinnedby periodic boundary conditions.

The distances between the diffusing oxygen atom A and its firstnearest oxygen atom B or C at the transition state are shown inFig. 4(d) and (f). These distances are smaller in even-number-sizesupercells (2.84 Å in 2 � 2 � 2 and 2.89 Å in 4 � 4 � 4), and largerin odd-number-size supercells (2.94 Å in 3 � 3 � 3 and 2.95 Å in5 � 5 � 5). Therefore, one can infer that the repulsive Coulombinteraction between the diffusing oxygen atom and its nearest oxy-gen atom in a 3 � 3 � 3 supercell is lower than that in 4 � 4 � 4supercell. As such, this weaker repulsive interaction wouldenhance the oxygen diffusion rates by lowering diffusion barriers.This difference in the local atomistic environment in differentsupercells, due to the artificial boundary conditions, is consideredto be the cause of the observed anomaly and explains why the

Fig. 6. Charge density difference of saddle point structure between containing charged oSTO supercells. In these figures, green, red, and blue balls represent strontium, oxygen anvalues are shown in light blue and yellow. The primitive unit cell containing the vacancyfigure legend, the reader is referred to the web version of this article.)

MEB in a 3 � 3 � 3 supercell is lower than that in the 4 � 4 � 4supercell.

3.3.2. The effect of charge states of the oxygen vacancyWe further examine the effect of charge states of the oxygen

vacancy on the MEB. Results using PBEsol are shown in Fig. 5.The MEB of a neutral oxygen vacancy is found to be higher thanthat of a charged vacancy, which is consistent with the previousstudy carried out by Al-Hamadany et al. [23]. The MEB differencebetween the neutral and charged vacancies is substantial (as highas 0.35 eV) in small supercells (2 � 2 � 2). However, this energydifference decreases quickly to less than 0.05 eV upon increasingsupercell size.

To further elucidate the effects of charge states on oxygenvacancies, the charge density differences of the saddle point con-figurations between charged and neutral states are analyzed, asshown in Fig. 6. For consistency, charge density difference calcula-tions using the atomic positions of the relaxed neutral configura-tions as a reference structure were employed to obtain inducedcharge density distributions for both the neutral and chargedvacancy. We observe that with increasing supercell size, the excesscharge (2e�) becomes more delocalized, reducing the total chargedensity on titanium and oxygen atoms, thereby diminishing theeffect of the extra charge on MEBs. This is indeed more pronouncedin larger supercells (n � n � n, n > 3) where there is a relativelysmaller difference in the MEBs for neutral and charged vacancies.

3.3.3. The effect of +UAs mentioned above, the charge density should be strongly

localized around the Ti atoms adjacent to the oxygen vacancy.The GGA + U method adds a Hubbard-type term to the densityfunctional that penalizes partial occupancies in correlated orbitalsand subsequently increases electron localization. Fig. 7 shows themigration energy barrier of (a) a neutral oxygen vacancy and (b)a charged oxygen vacancy calculated using either the PBEsol orPBEsol + U functional. In general, we find the inclusion of aHubbard-U type term results in higher MEBs for neutral vacancies(the exception being the 2 � 2 � 2 case). For a charged oxygenvacancy in the large 5 � 5 � 5 supercell, we obtain a MEB of0.40 eV, 0.48 eV, 0.33 eV, and 0.27 eV using PBEsol + U, PBE + U,PBEsol, and PBE, respectively (see Table 3).

xygen vacancy and neutral vacancy in 3 � 3 � 3 (a), 4 � 4 � 4 (b), and 5 � 5 � 5 (c)d titanium atoms; the isosurface value is 0.00018 e/Bohr3, the negative and positiveis indicated by dashed red box. (For interpretation of the references to color in this

Fig. 7. Migration energy barriers of (a) a neutral oxygen vacancy and (b) a chargedoxygen vacancy calculated with the PBEsol and PBEsol + U functionals.

Table 3Diffusion energy barriers of an isolated oxygen vacancy in STO supercells calculatedwith different DFT methods.

DFTmethod

Vacancy chargestate

Energy barrier (eV)

2 � 2 � 2 3 � 3 � 3 4 � 4 � 4 5 � 5 � 5

SP_PBE V0 0.81 0.34 0.32 0.30V2+ 0.44 0.33 0.30 0.27

PBE + U V0 1.62 0.82 0.61 0.49V2+ 0.65 0.54 0.58 0.48

PBEsol V0 0.90 0.43 0.47 0.39V2+ 0.55 0.41 0.43 0.34

SP_PBEsol V0 0.89 0.42 0.46 0.38V2+ 0.54 0.40 0.43 0.33

PBEsol+ U

V0 0.82 0.50 0.64 0.39V2+ 0.69 0.55 0.64 0.40

Note: SP = spin polarized.

314 L. Zhang et al. / Computational Materials Science 118 (2016) 309–315

Applying Hubbard-type terms in DFT calculations can signifi-cantly affect the charge density distribution of the STO supercells.Fig. S1 (supplementary material) shows the charge density differ-ence between PBEsol + U and PBEsol for saddle points in differentSTO supercells containing a neutral oxygen vacancy. We observethat the charge density difference is localized to regions surround-

ing the Ti atoms due to the Hubbard-type term in DFT making theelectronic Ti d-orbitals more localized. Because of the charge inter-action effect between titanium and its neighboring oxygen atoms,the orbitals of the neighboring oxygen atoms also become local-ized. An oxygen vacancy in an STO supercell significantly changesthe electronic states of atoms near the vacancy site. It is well estab-lished that localized states can be more adequately described withthe application of a Hubbard-U term, thus implying that DFT + Ucalculations may be more reliable for predicting MEBs for an oxy-gen vacancy in STO supercells. In addition, density of states (DOS)for the 3 � 3 � 3 supercell were calculated using the PBEsol + Uand PBEsol functionals. The valence band maximum (VBM) andconduction band minimum (CBM) gap calculated by PBEsol + U iscloser to the experimental value, seeing supplementary material.This also indicates that adding a Hubbard-type term can moreaccurately describe the electronic states of STO supercell and fur-ther predict a more reliable vacancy migration energy barriers.

4. Conclusion

The migration energy barrier of an isolated oxygen vacancy wassystematically examined using DFT. Our results indicate that arti-ficial boundary conditions, which are limited by finite supercellsize, are important causes of deviations in the calculations–typi-cally resulting in dramatic overestimations of migration energybarriers in small supercells. Increasing the size of the supercellemployed in the simulations leads to decreased values of theenergy barrier. When a sufficiently large supercell is employed, dif-ferent simulation approaches result in similar values, between 0.39and 0.49 eV. Limited by supercell boundary conditions, the octahe-dral rotation induced by oxygen vacancy generates two differentresults. One outcomemakes the distance between diffusing oxygenatom and its nearest oxygen longer (3 � 3 � 3 or 5 � 5 � 5 super-cells) and the other makes it shorter (2 � 2 � 2 or 4 � 4 � 4 super-cells), which affect the oxygen vacancy migration energy barrier indifferent size supercells. On the other hand, the computed MEBs ofneutral and charged oxygen vacancy indicate that a chargedvacancy has a lower MEB than the neutral vacancy and that theeffects of different DFT functionals and vacancy charge states aresubstantial in small 2 � 2 � 2 supercells and diminishing as super-cell size increases. Finally, a comparison of the energy barriers andinduced charge density distributions calculated with and withoutthe addition of a Hubbard-U term suggests that PBEsol + U andPBE + U methods can more accurately describe the electronic stateof STO supercells and thus would be preferable for studyingvacancy migration within this material. We note that our predictedvalues lie below the experimental value of �0.60 eV, probably dueto the limitation of DFT functional chosen and the complex defectinteractions in experimental samples. However, our work providesa deep analysis of DFT simulations of oxygen vacancy MEBs in STOsupercells. These results provide fundamental insights on oxygenvacancy diffusion in STO, which can be extended to all perovskitestructure materials.

Author contributions

LZ and HX directed the study, analyzed results and wrote themanuscript. BL and PK did part of the calculations, analyzed theresults. HZ, VRC, PG analyzed the results. All authors made contri-butions to write the manuscript.

Acknowledgements

This research is sponsored by The University of Tennessee (UT)Science Alliance Joint Directed Research and Development Program

L. Zhang et al. / Computational Materials Science 118 (2016) 309–315 315

(LZ and HX) and the Laboratory Directed Research and Develop-ment Program of Oak Ridge National Laboratory (VRC, HZ, PG,and PRCK), managed by UT-Battelle, LLC, for the US Departmentof Energy (DOE); this research used resources of The NationalInstitute for Computational Sciences at UT under contractUT-TENN0112 and the National Energy Research Scientific Com-puting Center, which is supported by the DOE Office of Scienceunder Contract No. DE-AC02-05CH11231.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2016.02.041.

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