Abstract
The electronic band gap is a fundamental material parameter requiring control for light harvesting, conversion and transport technologies, including photovoltaics, lasers and sensors. Although traditional methods to tune band gaps rely on chemical alloying, quantum size effects, lattice mismatch or superlattice formation, the spectral variation is often limited to <1 eV, unless marked changes to composition or structure occur. Here we report large band gap changes of up to 200% or ~2 eV without modifying chemical composition or use of epitaxial strain in the LaSrAlO_{4} RuddlesdenPopper oxide. Firstprinciples calculations show that ordering electrically charged [LaO]^{1+} and neutral [SrO]^{0} monoxide planes imposes internal electric fields in the layered oxides. These fields drive local atomic displacements and bond distortions that control the energy levels at the valence and conduction band edges, providing a path towards electronic structure engineering in complex oxides.
Introduction
In superlattices and heterostructures, the Coulombic interaction between adjacent atomic layers of (nominally) charged and/or neutral planes can create internal electric fields^{1,2,3} that induce or alter many functional electronic, ferroic and optical properties^{4,5,6,7,8,9}. In the majority of atomically layered oxides explored to date, however, the heterostructures are constructed by interleaving two or more bulk materials, often perovskite ‘blocks’ with welldefined threedimensional connectivity^{10,11,12,13,14,15}. Rarely are the starting oxides natural heterostructures themselves, for example, Aurivillius, DionJacobson or the RuddlesdenPopper (RP) phases. The latter (ABO_{3})_{n}/(AO) (also denoted as A_{n+1}B_{n}O_{3n+1}) RP structure has n ABO_{3} perovskite blocks stacked along the [001] direction with an extra sheet of AO rocksalt layers interleaved every n perovskite layers. This geometry disconnects the BO_{6} octahedra along one direction and imposes severe constraints on the nearestneighbour interactions^{16}, promoting anisotropy in the structurederived electronic, transport and magnetic properties, which contrasts sharply with the 3D perovskite analogues^{17,18,19,20}. Indeed, the RP structure dimensionality^{21,22}, that is, systematic stacking of the n layers along [001] direction, has recently been exploited^{23} in combination with epitaxial strain to achieve highly tunable lowloss dielectrics. Yet, the range over which band gap control may be achieved in oxide heterostructures derived from twodimensional building blocks remains less explored and largely unknown.
In n=1 RP oxides, there are three types of Acation arrangements that produce [001]cation ordering without changing the number of formula units per cell relative to the A_{2}BO_{4} aristotype^{24}. We denote the stoichiometry of the ordered variants as AA′BO_{4}, where A and A′ are two chemically distinct cations with 100% site occupancy. In the context of this work, A, A′ and B are La^{3+}, Sr^{2+} and Al^{3+} cations, respectively, and the corresponding n=1 RP oxide stoichiometry is denoted as LaSrAlO_{4}. The Acation ordering scheme produces the longrange periodic ordering of [LaO]^{1+} and [SrO]^{0} planes in the crystal structure as shown in Fig. 1. Normally, such longrange chemical orderings are experimentally detected using diffraction measurements through the appearance of superlattice reflections, which are forbidden in a fully disordered or solid solution phase. As there are three different ways to order [LaO]^{1+} and [SrO]^{0} planes without cell multiplication, we introduce two structural parameters, λ and ξ, that provide a measure for classifying the orderings, which have both the same composition and number of formula units. This scheme can also describe longer period sequences not directly examined herein, and is also useful to describe the interlayer interactions across disconnected BO_{6} units^{25}. λ and ξ capture the relative proximities of the positively charged [LaO]^{1+} and neutral [SrO]^{0} monoxide layers along [001] direction (see Fig. 1). λ defines the separation, quantified in terms of the number of metal oxide planes along [001] direction, between two chemically equivalent [LaO]^{1+} (or [SrO]^{0}) layers, and ξ defines the separation between two nearest chemically inequivalent [LaO]^{1+} and [SrO]^{0} layers with distinct charged states.
It is the delicate balance of electrostatic interactions among these layers that provides a handle to direct the Al–O bond distortions and ionicity in LaSrAlO_{4} through Asite ordering. The ratio of these parameters, η=ξ/λ, is used to distinguish each phase by layer sequence; a local structural description of how η distinguishes each variant is described below and emphasized by the colourcoding scheme in Fig. 1. Here, we seek to understand the effects of λ and ξ on the electronic band gap in LaSrAlO_{4} and predict its change on ordering. Experiments of the solid solution phase (η=0) report an indirect band gap (E_{g}) near ~4–5 eV (refs 26, 27, 28), whereas the proposed Asite ordered structures examined herein remain to be synthesized and are unknown.
More recently, we computationally demonstrated the importance of Asite cation ordering and η parameter on the rocksalt layers of n=1 RPs in metallic LaANiO_{4} with correlated electrons, where A=Ca^{2+}, Sr^{2+} and Ba^{2+} (ref. 25). The ordering sequence of Asite cations on (001) AO monoxide planes in LaANiO_{4} RP oxides alters the electrostatic interactions, through a socalled ‘electrostatic chemical strain’ (ECS) effect^{25}, and was used to manipulate the 3dorbital structure at the Fermi level. It is termed an ECS effect, because of the manner in which these artificial oxides are constructed. The unit cell is obtained by ordering nominally positively charged [LaO]^{1+} and neutral [AO]^{0+} metal oxides, thereby creating a continuously varying ‘electrostatic’ potential due to dipolar interactions derived from [LaO]^{1+} and [AO]^{0+} ordering not appearing in the solid solution material. Our density functional theory (DFT) calculations revealed that the bond stresses^{29} from the chemically imposed local electric fields are accommodated by changes to the bond lengths and bond angles. It is the change in bond length between an ordered and disordered (solid solution) system relative to the disordered system that leads to the local microscopic ‘strain’ within the ECS framework. Furthermore, the charge state of the ions and their relative sizes, which we refer to as chemistry of the element, were also identified to be important. Thus, the interplay of electrostatics, bond strain and chemistry manifests as an ECS in layered oxides with ordered cation sublattice.
In this work, we focus on a prototypical insulating RP oxide, LaSrAlO_{4}, and show that Acation ordering, without any change to stoichiometry, enables deterministic control over the electronic band gap of the material (E_{g}). We report that the arrangement of the AO monoxide [LaO]^{1+} and [SrO]^{0} layers within a singleunit cell directs the magnitude of E_{g} through internal electric fields, which drive bond distortions producing large reductions in the conduction band minimum: it varies by ~2 eV in magnitude over the monoxide layering sequences explored. Our work establishes heterovalent Acation ordering as a novel ‘control knob’ in addition to epitaxial strain and layer dimensionality (n) to design the properties of artificial complex oxides from modular building blocks that are naturally layered.
Results
Energetics
We obtain equilibrium LaSrAlO_{4} cation ordered structures by systematically exploring the energetics of various lattice modes obtained from computed phonon dispersion curves for each sequence. Our frozen phonon calculations indicate the high symmetry structures of all three LaSrAlO_{4} sequences are dynamically stable (see Supplementary Figs 1–3). Figure 1 depicts the ground state structures for the three Asite ordered variants. No AlO_{6} octahedral tilting or rotations occur, only bond elongations and rumpling distortions, that is, the Al and O atoms are displaced away from the AlO equatorial plane. On comparing the total energies for the three sequences, the η=2 sequence is found to be the most stable ordered variant (Fig. 1c), with the η=1/3 and 1/2 sequences 4 and 242 meV per formula unit (f.u.), respectively, higher in energy (see Supplementary Fig. 4).
Groundstate structures and dynamical charges
The groundstate structure for the η=1/3 sequence is polar (space group I4mm, Fig. 1a) owing to the Acation compositional order removing inversion symmetry^{24}. The lattice tetragonality (c/a) is 3.324, and the AlO_{6} octahedra have four crystallographically equivalent equatorial (eq) Al–O_{eq} and two unique (one short and one long) apical (ap) Al–O_{ap} bonds, which give rise to an electric polarization of 7.2 μC cm^{−2} along the [001] direction as obtained from the Berry phase method. The asymmetry in the Al–O_{ap} bond lengths arises from the negatively charged [AlO_{2}]^{1−} layer being interleaved between a positively charged [LaO]^{1+} and a neutral [SrO]^{0} layer. The O_{eq} atoms in the [AlO_{2}]^{1−} layer are attracted towards the positively charged [LaO]^{1+} layer, which simultaneously buckles the O_{eq}–Al–O_{eq} bonds so they deviate from 180°. Interestingly, we find no anomalously large or small Born effectivecharges Z* (relative to the nominal charge) in the ground state (Table 1); however, anisotropy between the inplane ( and ) and outofplane is evident.
In contrast, the η=1/2 LaSrAlO_{4} structure is centrosymmetric and nonpolar (space group P4/mmm). Figure 1b shows that, although the AO monoxide sequence splits the Al sites into two distinct AlO_{2} sublattices, we find that the c/a value (3.318) is close to that of η=1/3. Unlike the η=1/3 sequence, where the O_{eq}–Al–O_{eq} bond angles are <180^{°} and there are two Al–O_{ap} bonds, here there is no rumpling (∠O_{eq}–Al–O_{eq} =180°). At each site, the two Al–O_{ap} bonds are also of equal length; however, the relative lengths of the Al–O_{ap} bonds differ between the two AlO_{6} octahedra. In the [LaOAl(1)O_{2}LaO] and [SrOAl(2)O_{2}SrO] perovskitelike blocks, the Al–O_{ap} bonds are 1.96 and 2.12 Å, respectively, indicating a markedly different coordination and chemical environment owing to the cation order (Fig. 1b). Indeed, while the inplane (and ) for both Al sites is close to the nominal ionic charge of +3e, the outofplane components show significant deviations from the nominal value that depend on the adjacent monoxide layer chemistry. The values are 2.57e and 3.70e for Al(1) and Al(2), respectively, with the shorter Al–O_{ap} bond lengths reflecting stronger covalent interactions^{30} with the O(2)_{ap} atoms in the LaO layer. Note that the sign of the charge anisotropy ΔZ(La) also changes from negative (in η=1/3) to a positive value in this nonpolar variant, indicating that [LaO]^{1+} layers are more sensitive to cation ordering relative to the [SrO]^{0} layers (despite ξ being the same as before).
Similar to the η=1/3 phase, for the η=2 aluminate there are [AlO_{2}]^{1−} layers interleaved between [LaO]^{1+} and [SrO]^{0}, yet the structure is centrosymmetric (P4/mmm) due to antipolar displacements rather than symmetric stretching. The O_{eq} atoms shift towards the [LaO]^{1+} layers and a significant disproportionation of the Al–O_{ap} bonds occurs with large deviations of the O_{eq}–Al–O_{eq} bond angles away from 180°. The electrostatic effects within the unit cell are further enhanced, because the two [LaO]^{1+} layers are in close proximity to one another, which leads to a repulsive interaction. The [LaO]^{1+}–[LaO]^{1+} repulsion is evident in the asymmetry of the Al–O_{ap} bond lengths, where we find 2.37 and 1.85 Å for the O_{ap} oxygen in the [LaO]^{1+} and [SrO]^{0} layers, respectively. The overall consequence of the enhanced electrostatic effects are that the η=2 LaSrAlO_{4} structure has the largest c/a axial ratio (3.44), compared with other ordered variants, including the random alloy (3.36)^{31}. Except for a large for the O(3)_{ap} atoms found in the [SrO]^{0} layers (due to the shorter Al–O(3)_{ap} bond length), the Z* for the remaining ions are closer to the nominal values. It is also worth noting that the magnitude of the anisotropy ΔZ for La and Sr is greater in the η=2 sequence relative to the η=1/3 sequence (Table 1). However, we find ΔZ(La)< 0, similar to η=1/3.
Internal electric fields
In Fig. 2, we show the macroscopically averaged^{32,33} local nuclear ionic potential, excluding contributions of the exchangecorrelation and Hartree terms, along the [001] direction obtained from our firstprinciples calculations for each sequence. The builtin internal electric fields, which are proportional to the gradient in the potential, are the source for the cationorderingdriven bondlength modulations. The dipolar interactions provide the driving force for the cooperative atomic displacements (as shown in Fig. 1), which occur to minimize the interatomic forces for a given η in LaSrAlO_{4}.
Figure 2a,c clearly depicts the different electric fields that occur across the AlO_{2} layers in the η=1/3 and 2 sequences (arrowed). The steeper slope produces a larger asymmetry in the Al–O_{ap} bonds, that is, the magnitude of local electric field across the AlO_{2} layers is greater for η=2 than in the η=1/3 sequence. In addition, the field does not cancel in η=1/3 owing to the polar crystal structure. Interestingly, in the η=1/2 sequence the macroscopic average profile of the two perovskitelike blocks, [SrOAlO_{2}SrO] and [LaOAlO_{2}LaO] are markedly different (Fig. 2b). The local minimum in the macroscopic averages do not coincide in η=1/2 (for example, see filled symbols indicating Al sites), unlike the other two sequences. We anticipate these variations in the internal electric field from cation ordering to alter the positions of the valence and conduction band edges.
Electronic structure and band gaps
Figure 3 shows the electronic band structures obtained from our DFTPBEsol simulations for the three cationordered sequences. All three configurations are found to be insulating, with the top of the valence band (VB) and bottom of the conduction band (CB) formed mainly from O 2p and La 5d states, respectively. Note that the bandwidths of the O 2p and La 5d states are also altered as a function of η, but more remarkably, the magnitude of the indirect band gap (E_{g}, indicated by the blue arrows) is significantly affected by the AO layering sequence.
The polar η=1/3 LaSrAlO_{4} structure has the largest E_{g} (3.0 eV), followed by the η=2 sequence, which is slightly reduced (2.8 eV). Moreover, the η=1/2 structure with two distinct A and Bsite sublattices has a significantly reduced E_{g} of 1 eV. Note that the inclusion of the La 4fstates in the pseudopotentials increased the E_{g} of η=1/2 by 0.5 eV, but has a negligible effect on the electronic gaps of η=1/3 and 2 sequences. For η=1/2 LaSrAlO_{4}, we find two important changes to the band edges that are notably absent in both η=1/3 and 2 sequences. Figure 3b shows a shift in the band edges and an enhanced dispersion of both the O 2p states from Γ–M–X, originating from the O(4)_{eq} atoms in the [SrOAlO_{2}SrO] block in the VB, and the La 5d states at the bottom of the CB. The combination of these two effects, which are less pronounced in η=1/3 and 2, leads to the massive 2 eV band gap reduction. For comparison, we also report the E_{g} values obtained using an allelectron DFT scheme in Table 2 (see the row labelled Wien2k and the Methods section for details). Although the band gap values of the η=2 and η=1/3 configurations using the full potential method remain essentially unaltered relative to the ultrasoft pseudopotential values (see Table 2), the inclusion of the La 4f states increases the band gap by 0.6 eV in η=1/2. This finding is consistent with results using pseudopotentials that explicitly treat the 4f states described above.
Our calculations with the HSE06 functional, which includes a fractional contribution of exactFock exchange, and with the Green’s function G_{0}W_{0} method are known to more accurately reproduce band gaps^{34,35} also find a similar 2.3 eV reduction for the η=1/2; albeit, as expected, the absolute value of the gap differs between methods (Table 2). The direct Γ→Γ band gaps also show similar behaviour, with that the η=1/2 sequence considerably smaller than that of the η=1/3 and 2 sequences.
Origin of gap variation
To identify the atomic features responsible for the band gap reduction, we probe the sensitivity of the electronic structure to various lattice distortions appearing in the equilibrium η=1/3, 1/2 and 2 structures. We performed a set of computations starting with the experimental structure of the η=0 LaSrAlO_{4} solid solution, where the La and Sr cations are randomly distributed. The η=0 LaSrAlO_{4} belongs to I4/mmm space group and its lattice constants a and c are 3.754 and 12.64 Å, respectively. With the η=0 structure, we ordered La and Sr on the Asites corresponding to the η=1/3, 1/2 and 2 sequences.
We next performed selfconsistent DFT calculations using ultrasoft pseudopotentials at the PBEsol level of theory, where we neither relaxed the internal coordinates nor the unit cell geometry and calculated the electronic band gap. In this way, we are able to isolate the change in the electronic structure independent of the induced lattice and bond distortions to identify the [LaO]^{1+} and [SrO]^{0} ordering effect on the band gap. We found that the E_{g} of the η=1/3 and 2 sequences reduced from 3.0 and 2.8 eV, respectively in the ground state (see Table 2) to values of 2.77 and 0.16 eV, respectively. In contrast, the η=1/2 sequence became metallic. These calculations when compared with the results obtained from allowing differential ionic relaxations due to the chemical ordering reveal two key effects: (i) the cation ordering drives large band gap variations to such an extent that the η=1/2 sequence becomes metallic and (ii) the critical role of lattice relaxations, that is, variation in the bond lengths (strains) and angles, in response to the cation ordering sequences restores the gap. Our findings highlight the important interplay of electrostatics and local bond strain by [LaO]^{1+} and [SrO]^{0} ordering to tune the electronic properties of LaSrAlO_{4}. The cation order drives a reduction in the E_{g} from the large electric fields (as shown in Fig. 2), whereas the screening provided by differential atomic relaxations^{36} reduces these field strengths and maintains the insulating behaviour of LaSrAlO_{4}. As the variation in the fields for each sequence is different (due to different η values), we arrive at the variation in the E_{g} (Table 2).
Furthermore, we also explored the sensitivity of the conduction band edge, (that is, La 5d states) for the η=1/2 sequence with the smallest E_{g}. We cooperatively displaced the two O_{ap} atoms in the [LaOAlO_{2}LaO] block, such that P4/mmm symmetry is preserved. Supplementary Fig. 5 shows that as the two Al–O_{ap} bond lengths decrease (increase) relative to the ground state configuration, the energy difference between the minimum in the macroscopically averaged local ionic potential between the two perovskitelike blocks increases (decreases). This bond contraction (elongation) results in an increase (decrease) of the indirect band gap (Supplementary Fig. 5b–d.) As the induced Al–O displacements are produced by the buckling of the LaO monoxide layers in response to the electrostatic effect, our computational exercises conclusively show that the E_{g} changes arises directly from the ordering of [LaO]^{1+} and [SrO]^{0} monoxide layers in the [001] direction—the cationorderingdriven bond length modulations provide a direct route to tune band structure using internal electric fields without elemental substitution.
Discussion
Using this principal finding of massive variations in the E_{g} trend without any changes to the stoichiometry through AO monoxide layering induced internal fields, we establish cation ordering as a novel means for electronic structure engineering in materials with welldefined twodimensional planes. As transition metal oxides exhibit many functional properties, including anisotropic transport, strong electronlattice coupling (superconductivity and magnetoresistance), relative ease of ionexchange and intercalation, low thermal conductivity and ferroelectricity (to name a few), this understanding could lead to layered oxides as a platform for multilfunctional and adaptive electronic materials.
To realize our theoretical predictions, we suggest thin film growth methods like oxide molecular beam epitaxy (MBE) to be wellsuited for experimentally achieving the proposed cation ordering variants^{37,38,39}. Recently, Asite cation ordered LaSrNiO_{4} and LaSrMnO_{4} RP oxides were experimentally grown with precise atomic control using oxideMBE techniques^{40,41}. Leveraging, this approach with misfit strain from epitaxial growth or by tuning the magnitude of the cationorderingdriven bond length modulations via large Asite valence differences offers a potentially transformative paradigm to rationally design band gaps, absorption and emission edges, or transport properties for photo and electrocatalysis, where such layered RuddlesdenPopper and DionJacobson phases are already finding use, yet remains far from ideal.
Methods
Electronic structure calculations
Density functional theory calculations were performed within the generalized gradient approximation (GGA) PBEsol exchange—correlation functional^{42} as implemented in the planewave pseudopotential code, Quantum ESPRESSO (QE)^{43}. In calculations involving QE, the core and valence electrons were treated with ultrasoft pseudopotentials^{44} with the inclusion of nonlinear core corrections with a planewave basis (60 Ry cutoff). See Supplementary Note 1 and Supplementary Table 1 for additional calculation details. The following valence electronic configurations were used for our TroullierMartin^{45} pseudopotentials, La: 5s^{2}5p^{6}5d^{1}6s^{2}; Sr: 4s^{2}4p^{6}5s^{2}; Al: 3s^{2}3p^{1}; and O: 2s^{2}2p^{4}. The Brillouin zone integration was performed over a 8 × 8 × 5 MonkhorstPack kpoint mesh^{46} centred at Γ. The atomic positions and the cell volume were relaxed until the HellmannFeynman forces and stress tensor <2 meV Å^{−1} and 0.1 kbar, respectively, were achieved.
Lattice dynamical calculations were performed using PHONOPY^{47} on a 2 × 2 × 2 supercell with forces obtained from selfconsistent total energy calculations using the finite displacement method (see Supplementary Note 2). The electric polarization (ΔP) for the polar LaSrAlO_{4} variant is calculated using the Berry phase method^{48}. The Born effectivecharge tensors (Z*), which characterize the influence of longrange Coulombic interactions, were calculated using the density functional perturbation theory (DFPT) for the equilibrium structures^{49}.
We also used the allelectron code Wien2k^{50} to validate the suitability of our pseudopotentials for accurately describing the electronic band structure. Wien2k calculations were also performed within the PBEsol exchangecorrelation functional. The muffintin sphere radii were chosen as follows; La: 2.2, Sr: 2.0, Al: 1.7, O: 1.6 a.u. and we used a planewave cutoff of RK_{MAX} =7. The Brillouin zone integration was performed over a 14 × 14 × 4 MonkhorstPack kpoint mesh^{46} centred at Γ. We relaxed the lattice using MSR1a for optimization of positions^{51}. Note that the relaxed lattice and internal coordinates from Wiek2k agreed well with the Quantum ESPRESSO simulations.
Given the inherent limitation in the DFT exchangecorrelation functionals in accurately predicting band gaps, we also used the hybrid exchangecorrelation functional HSEsol^{52}, and a beyondDFT (G_{0}W_{0}) method as implemented in the BerkeleyGW^{53,54,55} code to substantiate the band gap variations observed with changes in η. Normconserving pseudopotentials constructed using the OPIUM pseudopotential package ( http://opium.sourceforge.net/) were used for the HSEsol and G_{0}W_{0} calculations.
DFT calculations were performed on a 8 × 8 × 5 mesh to obtain wellconverged singleelectron wave functions using the GGAPBEsol functional as a starting guess for calculating the quasiparticle energies and wavefunctions within the G_{0}W_{0} method. The process of computing the quasiparticle energies requires two steps: first, calculation of the static or frequencydependent polarizabilities (χ) and dielectric functions within the random phase approximation (RPA). The generalized plasmonpole model is used to extend the dielectric function to finite frequency. We used a dielectric cutoff energy of 50 Ry to solve for χ, from which the RPA dielectric matrix is calculated. Second, calculation of the screened Coulomb interaction, W, and the oneparticle Green’s function, G, from the inverse dielectric matrix (obtained from Step 1) and DFT eigenfunctions, respectively. These quantities are then used to evaluate the selfenergy operator (Σ) and obtain the quasiparticle energies. We used screened coulomb and bare coulomb cutoff values of 10 and 25 Ry, respectively. Additional details on the methodology can be found in the ref. 55.
For η=1/3 and 2 sequences, the complex flavour of the BerkeleyGW code was used. In contrast, we used the real flavour for the η=1/2 sequence. In Supplementary Table 3, the convergence of the indirect band gap (E_{g}) as a function of number of conduction or empty bands is given.
Additional information
How to cite this article: Balachandran, P. V. et al. Massive band gap variation in layered oxides through cation ordering. Nat. Commun. 6:6191 doi: 10.1038/ncomms7191 (2015).
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Acknowledgements
P.V.B. and J.M.R. acknowledge support from DARPA (grant no. N660011214224). P.V.B. acknowledges discussions with Dr A. Saxena, J. Young and N. Charles. We thank Professor S. Halasyamani, Professor B. NelsonCheeseman, Dr A. Bhattacharya and Dr J. Íñiguez for insightful conversations, and gratefully thank Professor L.D. Marks for performing the Wien2k calculations. P.V.B. and J.M.R. acknowledge the help of Dr J. Deslippe and Dr D. Strubbe with the BerkeleyGW code. The computational work made use of the GARNET and SPIRIT clusters at the ERDC and AFRL HPC resources, respectively, under the HPCMP initiative, and facilities available at the Center for Nanoscale Materials (CARBON Cluster) at Argonne National Laboratory, supported by the U.S. DOE, Office of Basic Energy Sciences (BES), DEAC0206CH11357. The views, opinions and/or findings reported here are solely those of the authors and do not represent any official views of DARPA.
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The study was planned, calculations carried out, and the manuscript prepared by P.V.B. and J.M.R. Both authors discussed the results, wrote, and commented on the manuscript.
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Balachandran, P., Rondinelli, J. Massive band gap variation in layered oxides through cation ordering. Nat Commun 6, 6191 (2015). https://doi.org/10.1038/ncomms7191
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