The role of screened exact exchange in accurately describing properties of transition metal oxides: Modeling defects in LaAlO3
Fedwa El-Mellouhi, Edward N. Brothers, Melissa J. Lucero, Gustavo E. Scuseria
UUPDATED: January 3, 2019
The role of screened exact exchange in accurately describingproperties of transition metal oxides: Modeling defects in LaAlO Fedwa El-Mellouhi,
1, 2, ∗ Edward N. Brothers, † Melissa J. Lucero, and Gustavo E. Scuseria
3, 4, 51
Chemistry Department, Texas A&M at Qatar,Texas A&M Engineering Building, Education City, Doha, Qatar Physics Department, Texas A&M at Qatar,Texas A&M Engineering Building, Education City, Doha, Qatar Department of Chemistry, Rice University, Houston, Texas 77005-1892 Department of Physics and Astronomy,Rice University, Houston, Texas 77005-1892 Chemistry Department, Faculty of Science,King Abdulaziz University, Jeddah 21589, Saudi Arabia
Abstract
The properties of many intrinsic defects in the wide band gap semiconductor LaAlO are stud-ied using the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE). As in pristinestructures, exact exchange included in the screened hybrid functional alleviates the band gap un-derestimation problem, which is common to semilocal functionals; this allows accurate predictionof defect properties. We propose correction-free defect energy levels for bulk LaAlO computedusing HSE that might serve as guide in the interpretation of photoluminescence experiments. PACS numbers: 71.15.Mb,71.15.Ap, a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov efects in LaAlO have been studied extensively both experimentally and using compu-tational approaches, contributing to our understanding of the interplay between variousdefects in this material. Photoluminescence (PL) spectroscopy using sub-band-gap exci-tation was recently used to detect the ground state defect states within the band gap ofLaAlO single crystals. In standard photoluminescence, electrons are pumped to the con-duction band then a photon is emitted upon relaxation from conduction band to variousground state defect levels. The resulting PL peaks are then associated with defect levelsinside the gap. In sub-bandgap excitation, the photon energy is tuned to selectively probecertain defect levels revealing more detailed features. This experiment identified three dis-tinct PL peaks, each showing doublet splitting, that were localized 2 eV below the con-duction band minimum (CBM). Defect levels calculated using the the generalized gradientapproximation density functional of Perdew, Burke and Ernzerhof (PBE) and correctedwith the “scissor operator” were used as a guideline to partially match the PL peaks. Thisapproach is less than completely satisfying, however, as (for example) the La A l defect level,post-correction, is located 1 eV below the CBM; this contradicts recent experimental results.A more accurate theoretical description is thus much needed, especially given the problemsof band gap underestimation (endemic to semilocal functionals) which is fatal for defectcalculations, and questions about the overall appropriateness of the “scissor operator.” Putmore simply, the typical theoretical methods which can be used for modeling these sorts ofmaterials are insufficiently accurate for explaining the effects in question.Defects in LaAlO have been subject to other very recent theoretical calculations: Vacancy defect energetics in rhombohedral and cubic bulk LaAlO have been computedusing PBE in Ref. 3, where it was found that the defect formation behavior in both phaseswere very similar. That work also included finite size scaling using supercells up to 480 atoms,suggesting that the cell-size dependencies in modeling neutral vacancies are almost negligible.(This makes their formation energies almost independent from the supercell size.) However,it should be noted that formation energies were modified using a band-gap correction scheme to overcome the well-known band gap underestimation problem of semilocal functionals.For this reason, interest has emerged in using modern (and demonstrably more accurate )screened hybrid functionals to model these defects. While some recent efforts have beenpublished in this direction, a complete picture of all possible defect levels using modernhybrid functionals is not available. 2n the present work, we apply the screened hybrid functional of Heyd-Scuseria-Ernzerhof(HSE) to a wide array of neutral defect types in LaAlO , thus complementing previousHSE efforts that treated only the oxygen vacancies. This work is motivated by HSE’sagreement with experiment for the calculation of many of the electronic, structural, andelastic properties in cubic LaAlO . HSE is expected to give point defect formation energiesand energy levels in close agreement with experiment as its direct and indirect band gaps as well as valence band widths (VBW) are in excellent agreement with experiment (seetable I); this can be contrasted with the PBE results, which have been previously used tostudy point defects in LaAlO . It is worth noting that HSE06 gives an excellent agreementwith the results of the global hybrid PBE0 for the case of the oxygen vacancy in SrTiO .This suggests that hybrid functionals belonging to the 25% HF exchange family like PBE0and HSE06 would yield very similar location of the defect level and the splitting of theconduction band minimum in the LaAlO case as well.Here we restrict our study to neutral defects to avoid introducing errors due to spuriouselectrostatic interactions, and the corrections associated with it. Nevertheless, performingHSE calculations with the high numerical accuracy settings detailed below remains quiteexpensive, thus precluding the use of the largest supercells. This is acceptable, however, asfinite size scaling and previous investigations using larger supercells have shown that theneutral defects considered here suffer least from finite size effects. Consequently, despite thelimited number of atoms that can be treated with HSE, this approach promises increasedphysical accuracy compared to the less expensive semilocal functionals.All calculations presented in this paper were performed using the development versionof the gaussian suite of programs, with the periodic boundary condition (PBC) codeused throughout. The Def2- series of Gaussian basis sets were optimized following ourprocedure described in Ref. 16 for bulk LaAlO . As in Ref. 16, we use the notation SZVPto differentiate these optimized PBC basis sets from the molecular Def2-SZVP basis sets.The functionals applied in this work include PBE and HSE. The use of HSE imposes limitations on the size of the supercell that can be efficientlycomputed and fully relaxed, so a LAO supercell of 2 × × k -point mesh of 6 × ×
6, including the Γ point. Also,we modeled a larger supercell of 2 × × k -points,in order to discuss the importance of defect self-interactions, and the effect of varying the3efect concentration on the electronic properties of LAO.Most numerical settings in gaussian were left at the default values, e.g. , geometryoptimization settings, integral cut-offs, k -point meshes and SCF convergence thresholds.Unless otherwise noted, crystal structures used in the chemical potential calculations onLa, Al, Al O , La O were downloaded as CIF files from the ICSD, and then fully re-laxed/optimized. Isolated, neutral vacancies were introduced to the crystal structure ofcubic LAO by removing one atom of either O, La or Al, while La and Al antisites occupiedthe crystalline position. All structures containing the above defects were then fully relaxedusing HSE06. In order to avoid imposing a certain oxygen interstitial position, the oxygenatom was inserted far from the well-known interstitial sites followed with relaxation to thenearest minimum. At this point, we cannot be completely sure whether the configuration weobtained has the lowest formation energy; only a full energy landscape exploration methodcan reveal that.The calculations of neutral defect formation energies used the formalism of Zhang andNorthrup, namely the equation: E f = E T − [ E T (perfect) − n La µ La − n Al µ Al − n O µ O ] (1)where E T and E T (perfect) are the calculated total energies of the supercells containing thepoint defect and the perfect bulk host materials, respectively. The number of each elementremoved from the perfect supercell is represented by n x , while µ x corresponds to the atomicchemical potentials in an LaAlO crystal. Assuming that LaAlO is always stable, thechemical potentials of the these elements can vary in the following correlation: µ La + µ Al + 3 µ O = µ bulkLaAlO (2)Obviously, atomic chemical potentials are determined by the sample composition andcannot be ascertained exactly. However, they can be varied to cover the whole phase diagramof LAO splitting into Al O and La O bulk phases. Hence the calculated formation energiesfor the neutral point defects vary according to equilibrium positions such as “O-rich” and“O-poor” conditions.The calculated enthalpies of formation in idealized materials (non-relaxed structures) forphases containing La, Al and O are summarized in Table I and are compared to previous4 ABLE I. Comparison of calculated fundamental electronic properties of bulk cubic LaAlO fromthis work and previous studies. VBW stands for the valence band width. Calculated enthalpies offormation in eV/atom for idealized materials with phases containing La, Al and O are comparedto previous PBE calculations and experiment.This Work Previous WorkHSE PBE PBE Exp.Direct gap (eV) 5.0 3.54 - -Indirect gap (eV) 4.74 3.26 3.1 e -VBW (R → R)(eV) 8.00 7.50 - -∆H fAl O a b ∆H fLa O a c ∆H fLaAlO a d a Ref. 2 b Ref. 19 c Ref. 20 d Ref. 21 e Ref 9 calculations and experiments. As a general trend, the formation enthalpies computed withHSE are close to the results from semilocal functionals like PBE (this work), althoughthe HSE values are slightly higher. The only exception is LaAlO , where PBE tends tooverestimate the formation enthalpies and exceed the HSE value.The formation energies of defects in LAO as function of its composition are plotted inFigure 1. Under oxidizing conditions (points A and B) we identify the oxygen interstitial(O I ) as having the lowest formation energy; this is contrary to previous PBE results which5 B C D E05101520 F o r m a t i onene r g y ( e V ) V O V Al V La O I Al La La Al FIG. 1. Defect formation energies of isolated defects in cubic LAO computed using HSE at eachequilibrium point based upon the phase diagram in Ref. 2. predicted O I to be less stable than V La and other vacancy complexes. It is worth noting thatwe introduced the oxygen atom at a random position in the supercell avoiding well-knowninterstitial sites followed by a full relaxation of the system. The resulting configurationconsists of a 110 split interstitial (dumbbell) with an O-O bond of 1.38 ˚A. Since Luo et al. ; did not report their interstitial configuration, we could assume that our differences arisefrom different interstitial sites considered rather than computational.Focusing specifically at point A, V La is the next most stable defect. Our formation energyis about 3 eV higher than previously published results obtained using the PBE functionalin rhombohedral and cubic LAO. In terms of competition between V La and V Al , we find(using HSE) the same behavior seen using PBE in Ref. 1 and 2. Next in order of stabilityis V Al and Al L a with equal formation energies at point A, followed by V O , a behavior not6 IG. 2. Band structures and PDOS calculated with HSE/SZVP for the 2 × × supercellcontaining intrinsic defects. The top figures represent O I and V La introducing bands with a valenceband character. Al L a and V O (middle row) have bands above the mid gap. The bottom rowcontains La Al and V Al having defect bands below mid gap. The Fermi energy E F is indicated bya solid black line. The red bands indicate the occupied defect bands while the unoccupied defectbands are shown in blue. V Al CBM E ne r g y ( e V ) V O Al La VBMO I V La La Al HSE
CBM E ne r g y ( e V ) V Al La VBMO I La Al V Al PBE
FIG. 3. Schematic representation of the average location of the defect bands in the band gap ofLAO calculated with HSE/SZVP (left) and PBE from Ref. 2 (right) shifted using a scissor operator.Numbers in gray boxes refer to the location of the defect bands with respect to the valence bandmaximum (VBM). The dashed line refer to the mid gap. reported previously.
Moving from point A to point B, the order of increasing stability of defect types remainunchanged, except for Al L a , which has become less stable than V O . We report a formationenergy of 8 eV for V O , which is in excellent agreement with a recently computed HSE valueof 8.3 eV in rhombohedral LAO using a supercell of up to 135 atoms; note that in thisstudy other vacancy types and substitutions were not modeledUnder reducing conditions (point C, D, E), V O dominates the spectrum, in qualitativeand quantitative agreement with previous uncorrected PBE calculations , having an averageformation energy of 1.3 eV. The formation energy of V O calculated with HSE is lowered by0.1 eV when the supercell size increases from 40 to 90 atoms. Although not negligible, thisremains smaller than the differences reported in the charged states which are due to boththe strong elastic and electrostatic self-defect interactions. Obviously, calculations usinglarger fully relaxed supercells are required to determine at what size defect self-interactions(elastic effects) become negligible.Our results do not agree, however, with the recent formation energies computed by Ya-mamoto et al. who applied a band gap correction (a 2.48 eV shift) to the PBE formationenergies of V O . Applying the band gap correction in this case led to the conclusion that8chottky-type vacancy complexes are more stable than V O . We believe this to be an artifactof the correction they applied.It should be noted that interstitials like La I and Al I are not addressed in the presentstudy because their neutral charge state was not identified to be stable according to thePBE calculation of Luo et al. Also, our formation energy spectrum computed with HSEreveal that they exhibit very high formation energies.The various defects we will first discuss induce changes to the electronic properties ofcubic LaAlO , introducing defect levels within the band gap and/or lifting the degeneracyof the CBM and VBM as shown in figure 2.The oxygen split interstitial configuration (O I ), which is the most stable under oxidizingconditions, induces a strong distortion to the lattice, which in turn significantly impactsthe electronic structure. The CBM splits at Γ point by 440 meV, while the VBM alsosplits into three distinct bands. The fully occupied defect band composed of O 2 p statesis located on average at 0.13 eV above the VBM, has valence band character, and inducesa gap of 4.66 eV. V La , the second most stable defect under oxidizing conditions, createsthree empty non-degenerate valence bands, dominated by O 2 p orbitals originating from theO dangling bonds. Both HSE and PBE agree about the nature and the location of thesebands. However, our O I level is shallower than the previously reported PBE results, whichis probably due to differences in the interstitial configuration.Next to be evaluated are defects having in-gap states, namely Al L a , V O , La A l and V Al ,which show a localized electronic density around the defect region. The Al L a antisite defectmight play a role under oxidizing conditions due to its relatively low formation energy. WithHSE, we find that it induces an empty defect band in the gap at 2.93 eV above the VBM and2.0 eV below the CBM. This band might become populated upon doping or under excitation,and is dominated by O 2 p and Al s orbitals ( q.v. the PDOS). The bulk degeneracy of theVBM and CBM are not affected, and remain 3 and 2 fold degenerate, respectively. This isan indication that this defect does not introduce noticeable distortion or octahedral rotationinto the lattice, which is further confirmed by a structural analysis. However, using PBEwe find that the Al L a defect band is located at 1.22 eV above the VBM and 2 eV below theCBM, which is well below the mid gap (1.6 eV). Following a typical band gap correctionprocedure, this PBE defect band does not need to be shifted using the scissor operator, which would result in keeping its VB character, which contradicts the HSE results above.9he next defect of interest is V O , which is arguably the most important defect underreducing conditions, and suspected to be systematically introduced during the growth ofmetal oxide superlattices. After introducing V O , the supercell shrinks along the y axis,leading to a tetragonal distortion of the lattice with a ratio a/b =1.0057 ( a and b being thenew lattice parameters) and a slight rotation of AlO octahedra. This impacts strongly theelectronic structure by splitting the doubly degenerate CBM by 258 meV, while leaving theVBM triply degenerate. A new defect band also appears at 2.77 eV above the VBM fromthe combination of O 2 p , Al d , La d and p orbitals. Here again, major differences withprevious PBE data emerges: the uncorrected PBE level computed recently by Chen et al. was located 2.23 eV above the VBM. Luo et al. applied the scissor operator to this defectlevel, predicting it to lie at about 3.8 eV above the VBM.Last to be examined is La A l , which in the neutral state would rarely form under eitheroxidizing or reducing conditions. It introduces a fully-occupied triply degenerate defect bandlocated 2.06 eV above the VBM and 2.60 eV below the CBM. However, the PBE defect levelis at that method’s mid gap, lying 1.6 eV from the VBM and CBM. If a scissor operatorwas to be used, one could argue that this level should be shifted, placing it as close as 1 eVto the CBM (see figure 3).To conclude, there are fundamental differences between our HSE defect level spectrumand the one published earlier using corrected PBE data regarding the nature of the defectbands (see figure 3). We believe these differences originate from the criterion used to judgewhether the “scissor operator” should be applied. For example, HSE finds that Al L a andV A l have defect bands near mid gap, thus removing the PBE’s prediction of valence bandcharacter, which were reported previously. The same issue leads to significant differencesin the conclusions regarding V O . Overall, our defect levels calculated with HSE lie 2 eVbelow the CBM (see figure 3), which is in better agreement with recent experiment . ThisHSE defect level spectrum we propose here is correction free, and may be used to interpretexperimental photoluminescence data which place defect levels at 3.1, 2.1 and 1.7 eV. ACKNOWLEDGMENTS
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