•Carriers in degenerate gapped compounds can cause spontaneous non-stoichiometry•Spontaneous non-stoichiometry can result in formation of ordered vacancy compounds•Different ordered vacancy compounds can be realized by controlling growth conditions•Controllable non-stoichiometry is the knob that tunes transparency and conductivity Degenerate gapped compounds, those with their Fermi levels inside the conduction band and sufficiently large “internal band gaps” below this Fermi level, are shown here to have negative formation enthalpy for cation vacancies, leading to spontaneous non-stoichiometry and formation of characteristic ordered vacancy compounds (OVCs), even at low temperature, in defiance of Daltonian stoichiometry. Selection during the growth of a specific OVC can be used to mitigate the notoriously difficult conflict between transparency and conductivity, with potential interests beyond of transparent conductors, including electrides and photocatalysts. This surprising prediction is validated with two key experimental findings: silver beta-alumina Ag3Al22O34.5 leeches out silver atoms upon any attempt to introduce electron carriers, whereas a number of different OVC structures stabilized by effective reconstructions are observed during the synthesis of the tetragonal tungsten bronze Ba3Nb5O15. We highlight a class of materials representing an exception to the Daltonian view that compounds maintain integer stoichiometry at low temperatures and use this behavior to select ordered vacancy compounds (OVCs) striking a wanted compromise between carrier concentration, transparency, and phase stability, crucial for transparent conductors (TCs). We show that carriers in the conduction band (CB) of degenerate gapped BaNbO3, Ca6Al7O16, and Ag3Al22O34 compounds can cause a self-regulating instability, whereby cation vacancies form exothermically because a fraction of the CB electrons decays into the hole states formed by such vacancies, and this electron-hole recombination offsets the positive energy associated with vacancy bond breaking. This Fermi level-induced spontaneous non-stoichiometry can lead to the formation of OVCs with different optoelectronic properties and stable in different ranges of chemical potentials. Thus, we demonstrate how a window of opportunity can be determined between opposing tendencies of transparency, conductivity, and stability to design TCs. We highlight a class of materials representing an exception to the Daltonian view that compounds maintain integer stoichiometry at low temperatures and use this behavior to select ordered vacancy compounds (OVCs) striking a wanted compromise between carrier concentration, transparency, and phase stability, crucial for transparent conductors (TCs). We show that carriers in the conduction band (CB) of degenerate gapped BaNbO3, Ca6Al7O16, and Ag3Al22O34 compounds can cause a self-regulating instability, whereby cation vacancies form exothermically because a fraction of the CB electrons decays into the hole states formed by such vacancies, and this electron-hole recombination offsets the positive energy associated with vacancy bond breaking. This Fermi level-induced spontaneous non-stoichiometry can lead to the formation of OVCs with different optoelectronic properties and stable in different ranges of chemical potentials. Thus, we demonstrate how a window of opportunity can be determined between opposing tendencies of transparency, conductivity, and stability to design TCs. The fact that compounds manifest integer ratios between component elements (the law of definite proportions1Dalton J. A New System of Chemical Philosophy. Vol. 1. W. Dawson, 1808Google Scholar) has been the cornerstone of our understanding of formal oxidation states (taking up integer values) and defect physics (showing that violation of integer ratios by formation of defects costs energy and is thus unlikely at low temperatures). This thinking of the Daltonides school (the paradigm of stoichiometry) stood in stark contrast with the non-stoichiometric Berthollides2Berthollet C.-L. Essai de statique chimique. Vol. 2. F. Didot, 1803Google Scholar school, who argued that compounds could possess a range of compositions entirely dependent on the starting synthetic conditions. We point out an interesting class of exceptions to the Daltonide universal understanding, whereby a degenerate but gapped compound with Fermi energy (EF) inside the conduction band (CB) and a large internal band gap (Egint) between the valence band maximum (VBM) and conduction band minimum (CBM) (as shown in Figure 1A ) could form a significant concentration of low-energy vacancies, violating the rule of integer stoichiometry. This understanding has an important implication on transparent conductors (TCs),3Ginley D.S. Hosono H. Paine D.C. Handbook of Transparent Conductors. Springer Science & Business Media, 2010Google Scholar those rare compounds in which the generally mutually exclusive properties of optical transparency (usually common only in electrical insulators) and conductivity (usually common only in opaque metals) coexist. This internal contradiction has been the reason why finding good TCs has proven to be so difficult. The old class of TCs was developed by starting with an insulator (such as In2O3 or ZnO, schematically shown in Figure 1B) and then attempting heavy doping (by Sn or Al, respectively), making it conductive. This has faced severe “doping bottlenecks,”4Zhang S.B. Wei S.H. Zunger A. Overcoming doping bottlenecks in semiconductors and wide-gap materials.Phys. B Condens. Matter. 1999; 273-274: 976-980Crossref Scopus (61) Google Scholar whereby intended doping by electrons creates “electron killers” in the form of intrinsic acceptors. An alternative strategy5Zhang X. Zhang L. Perkins J.D. Zunger A. Intrinsic transparent conductors without doping.Phys. Rev. Lett. 2015; 115: 176602Crossref PubMed Scopus (31) Google Scholar is to start from a metal (Figure 1A) and attempt to make it transparent, avoiding doping bottlenecks. The deeper understanding of spontaneous non-stoichiometry discussed here clarifies that the latter approach can present an unusual window of opportunity, whereby (1) transparency, (2) conductivity, and (3) phase stability can coexist and be selected by zooming in on specific growth conditions (chemical potentials). Specifically, a compound that has the Fermi level inside the CB (a nominal band conductor with free electrons) could become opaque because too many electrons create a strong plasma absorption in the visible range. But if this compound also has a sufficiently large internal band gap below the Fermi energy (i.e., being degenerate but gapped, Figure 1A), it will create spontaneous defects (here, cation vacancies that are hole-producing acceptors) that, while violating Daltonian stoichiometry, also regulate the carrier concentration by compensating the native electrons via the spontaneously produced holes. It is the exothermic electron-hole compensation reaction that drives non-stoichiometry. Unlike the case of the limited formation of dilute vacancies in ordinary insulators (Figure 1B), if there is an internal gap (Figure 1A), then, at the concentrated limit, such vacancies can condense into ordered crystallographic arrays (Figures 2A and 2B), thus explaining the hitherto peculiar occurrence6Belsky A. Hellenbrandt M. Karen V.L. Luksch P. New developments in the inorganic crystal structure database (ICSD): accessibility in support of materials research and design.Acta Crystallogr. B. 2002; 58: 364-369Crossref PubMed Scopus (831) Google Scholar of macroscopically observed sequences of ground state ordered vacancy compounds (OVCs),7Hart G.L.W. Zunger A. Origins of nonstoichiometry and vacancy ordering in Sc1-x□xS.Phys. Rev. Lett. 2001; 87: 275508Crossref PubMed Scopus (3) Google Scholar, 8Zhang S.B. Wei S.-H. Zunger A. Stabilization of ternary compounds via ordered arrays of defect pairs.Phys. Rev. Lett. 1997; 78: 4059-4062Crossref Scopus (296) Google Scholar, 9Anand S. Xia K. Zhu T. Wolverton C. Snyder G.J. Temperature dependent n-type self doping in nominally 19-electron half-Heusler thermoelectric materials.Adv. Energy Mater. 2018; 8: 1801409Crossref Scopus (44) Google Scholar, 10Anand S. Xia K. Hegde V.I. Aydemir U. Kocevski V. Zhu T. Wolverton C. Snyder G.J. A valence balanced rule for discovery of 18-electron half-Heuslers with defects.Energy Environ. Sci. 2018; 11: 1480-1488Crossref Google Scholar such as BalNbmOn, with l:m:n ratios of 1:2:6, 3:5:15, 5:4:15, 7:6:21, 7:8:24, 9:10:30, and 26:27:81. We highlight cases in which the vacancy ordering is clearly visible (Figure 2B) versus cases in which, following vacancy ordering, further total energy relaxation yielded structural changes (reconstruction) that make it difficult to visualize the original vacancy ordering. These otherwise peculiar integer ratios emerge as stable, T = 0 K ground state structures obtained via a first-principles total energy search. It turns out that such cation vacancy compounds occur as a sequence of discrete phases, each with its own vacancy concentration (e.g., 7:8:24, 9:10:30, and 26:27:81, with 12.5%, 10%, and 3.7% of Ba vacancies, respectively) and each is stabilized at specific reactant chemical potentials (colored domains in Figure 2A). Because of the discrete vacancy ratios, each phase, as illustrated below, has its characteristic residual (post-compensation) electron concentration and optical response. Such phases (Figure 2A) could then, in principle, be selected during growth by targeting the requisite cation chemical potential, so that the desired optoelectronic properties can be achieved. This work demonstrates that controlled deviation from Daltonian stoichiometry can be used as a knob to regulate transparent conductivity. We validate our theory with two key experimental findings: silver β-alumina Ag3Al22O34.5 leeches out silver atoms forming Ag vacancies upon any attempt to introduce electron carriers, and tetragonal tungsten bronze Ba3Nb5O15, in which the cation vacancy formation favors the formation of secondary phases. Our paper is the first attempt to understand and unravel the intrinsic complex connection between stoichiometry and properties, creating a roadmap for the future. This work has broad implications, as degenerate gapped compounds have been generating increasing interest in many fields beyond that of TCs, including the colored metallic photocatalysts as seen in the substoichiometric Sr1−xNbO3,11Xu X. Randorn C. Efstathiou P. Irvine J.T.S. A red metallic oxide photocatalyst.Nat. Mater. 2012; 11: 595-598Crossref PubMed Scopus (378) Google Scholar the electron-donating promotors for catalysts12Kitano M. Inoue Y. Yamazaki Y. Inoue Y. Yamazaki Y. Hayashi F. Kanbara S. Matsuishi S. Yokoyama T. Kim S.W. Hara M. Hosono H. et al.Ammonia synthesis using a stable electride as an electron donor and reversible hydrogen store.Nat. Chem. 2012; 4: 934-940Crossref PubMed Scopus (864) Google Scholar to low work function compliance layers.13Hosono H. Kim J. Toda Y. Kamiya T. Watanabe S. Transparent amorphous oxide semiconductors for organic electronics: application to inverted OLEDs.Proc. Natl. Acad. Sci. USA. 2017; 114: 233-238Crossref PubMed Scopus (91) Google Scholar Furthermore, our work explains why many of these degenerate gapped compounds require exotic synthesis conditions to be prepared stoichiometrically. The formation of vacancies in ordinary insulators (Figure 1B) involves breaking of stable chemical bonds without restoring any of the spent energy, so a significant concentration of such defects can exist only at increased temperatures. The situation can be different in degenerate gapped compounds (with EF located at an energy ΔECB above the CBM, as in BaNbO3, Ca6Al7O16, and Ag3Al22O34 discussed below) with an internal band gap Egint between the CBM and VBM (Figure 1A). Here, the formation of dilute concentration of metal vacancies can create electron acceptor states near the valence band (at energy EDL), resulting in the opportunity for decay of q CB electrons into these electron acceptor states, thereby regaining the energy q(Egint + ΔECB − EDL), which reduces the vacancy formation energy accordingly (see Figure 1A). When this energy exceeds the bond-breaking energy, one expects spontaneous non-stoichiometry. This defines an electronic Fermi level mechanism for Berthollides2Berthollet C.-L. Essai de statique chimique. Vol. 2. F. Didot, 1803Google Scholar non-stoichiometry. To validate this concept, Figures 3A, 3C, and 3E show the results of density functional calculated formation energies of the Ba vacancy in BaNbO3, the Ca vacancy in Ca6Al7O16, and the Ag vacancy in Ag3Al22O34, as a function of the metal chemical potential (see Experimental Procedures). The allowed stable chemical potential regions (constructed by considering possible competing phases, see Experimental Procedures) of the respective bulk compounds are shown in Figures 3B, 3D, and 3F. We see that for the degenerate gapped compounds under cation-deficient chemical potential conditions, vacancy formation energies can be extremely low (in fact, negative). Whereas BaNbO3 and Ca6Al7O16 have stable chemical potential (green) zones at the respective stoichiometries indicated, Ag3Al22O34 does not. In fact, in the latter case, the Ag vacancy formation energy (Figure 3E) is so strongly negative under all chemical potential conditions, that the CB is empty and the parent degenerate Ag3Al22O34 phase is not stable (i.e., no green zone in Figure 3F). The negative formation energies of dilute vacancies open the possibility of vacancy condensation and long-range ordering (Figure 2B). To examine this possibility, we have calculated the T = 0 K stable phases (“ground state diagram” or “convex hull”) of such ternary structures. This entails searching for configuration versus composition that lies on the energy convex hull,14Ducastelle F. Order and Phase Stability in Alloys. North-Holland, 1991Google Scholar which defines the phases with energy lower than a linear combination of any competing phases at the corresponding compositions. We create candidate configurations by considering a base compound (BaNbO3, Ba3Nb5O15, Ca6Al7O16, or Ag3Al22O34), then create a replica of N such units of the base compound and add successively p metal vacancies, i.e., OVC = N × (base) + pVm, searching via total energy minimization for stable and metastable configurations. We also include experimentally known reconstructed OVCs, the compounds that satisfy the OVC expression but do not have clearly defined vacancy sites (e.g., Ba3Nb5O15, BaNb2O6, and Ba5Nb4O6). Available information on the experimental literature15Svensson G. Ba2Nb5O9 — an intergrowth of BaNbO3 (perovskite) and NbO.Mater. Res. Bull. 1988; 23: 437-446Crossref Scopus (38) Google Scholar, 16Svensson G. Werner P.-E. Determination of the composition of BaNbO3 using profile refinement and phase analysis.Mater. Res. Bull. 1990; 25: 9-14Crossref Scopus (17) Google Scholar, 17Hessen B. Sunshine S.A. Siegrist T. Jimenez R. Crystallization of reduced strontium and barium niobate perovskites from borate fluxes.Mater. Res. Bull. 1991; 26: 85-90Crossref Scopus (64) Google Scholar, 18Casais M.T. Alonso J.A. Rasines I. Hidalgo M.A. Preparation, neutron structural study and characterization of BaNbO3 - a Pauli-like metallic perovskite.Mater. Res. Bull. 1995; 30: 201-208Crossref Scopus (47) Google Scholar, 19Hessen B. Sunshine S.A. Siegrist T. Fiory A.T. Waszczak J.V. Structure and properties of reduced barium niobium oxide single-crystals obtained from borate fluxes.Chem. Mater. 1991; 3: 528-534Crossref Scopus (48) Google Scholar, 20Hwang Y.K. Kwon Y.-U. Syntheses and electrical properties of tetragonal tungsten bronze type solid solution Ba6−xLaxNb10O30+δ (x = 0, 1, 2, 3) and Sr6Nb10O30.Mater. Res. Bull. 1997; 32: 1495-1502Crossref Scopus (26) Google Scholar, 21D'yachenko O.G. Istomin S.Y. Fedotov M.M. Antipov E.V. Svensson G. Nygren M. Holm W. 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Neutron powder diffraction profile refinement studies on Ca11.3Al14O32.3 and CaCiO(D0.88H0.12).Acta Chem. Scand. A. 1987; 41: 110-112Crossref Google Scholar is provided in Figure 4, and the theoretical results of this work are summarized in Figure 5. The key point to notice is that there is a sequence of stable OVC phases, each having a specific concentration of electrons per formula unit (e/f.u.) in the CB (including possibly zero). This is illustrated in Figures 6A and 7A showing the ground state diagrams for Ba-Nb-O and Ca-Al-O systems and chemical potential stability diagrams (Figures 2A and 7B) demonstrating the phases that are stabilized as the chemical potentials of the atoms being removed are continuously changed between their allowed values. Finally, we show how a window of opportunity can be determined computationally between opposing tendencies of (1) stability (Figures 2A and 7B), (2) conductivity (Figures 6B–6D and 7C), and (3) transparency (Figure 8) to design new TCs.Figure 5Summary of the Computational Results on Stable OVCs in Ba-Nb-O and Ca-Al-O SystemsShow full captionThe OVCs are generated by creating supercells of a base cell as OVC = N × (base) + pVm, where N and p are integer numbers. Number of electrons (Ne) in the conduction band is given per formula unit (f.u.) of the base compound (i.e., BaNbO3 or Ca6Al7O16). In general, the number of electrons in the conduction band for the degenerate gapped Ca-Al-O and Ba-Nb-O compounds can be predicted from the sum of composition-weighted formal oxidation states assuming Ba2+, Ca2+, Al3+, Nb5+, and O2− for each material. ΔECB is the occupied energy range of conduction band and Egint is the internal gap between CBM and VBM (Figure 1A) estimated from the density of states. Types of materials I, II, and III stand for non-transparent metals, potentially TCs (shown in red), and insulators, respectively. If the OVC in the lowest energy state does not have clearly defined vacancy sites, it is labeled as reconstructed.View Large Image Figure ViewerDownload (PPT)Figure 6Stability and Electronic Structures of BaNbO3 and Its OVCsShow full caption(A) Convex hull for the Ba-Nb-O system indicating the OVCs in color. In Figure 3B, the OVCs were not included in the calculations of the stability green zone of BaNbO3, but they are included in (A) and Figure 2A. This redefines the chemical potential stability zone under which the base compound can exist, and consequently the formation energy of Ba vacancy in BaNbO3 is about 0 eV under the redefined cation-poor conditions.Density of states for (B) BaNbO3, (C) Ba6Nb10O30, and (D) BaNb2O6, indicating the number of electrons in the conduction band per formula unit.View Large Image Figure ViewerDownload (PPT)Figure 7Stability and Electronic Structures of Ca6Al7O16 and Its OVCsShow full caption(A) Convex hull for the Ca-Al-O system indicating the OVCs in color.(B) Chemical potential diagram for the Ca-Al-O system, showing the stability chemical potential zone for each stable OVC phase in colors corresponding to the ground states in (A). The gray zone corresponds to the prohibitive chemical potential stability zone of binary Ca-Al systems. In Figure 3D, the OVCs were not included in the calculations of the stability green zone of Ca6Al7O16, but they are included (A and B). This redefines the chemical potential stability zone under which the base compound can exist, and consequently the formation energy of Ca vacancy in Ca6Al7O16 is about 0 eV under the redefined cation-poor conditions.(C) Band structures for Ca6Al7O16 and its OVCs.View Large Image Figure ViewerDownload (PPT)Figure 8Effect of Non-stoichiometry on Optoelectronic PropertiesShow full caption(A) Schematic illustration of different contributions to optical properties in degenerate gapped compounds.(B) Absorption spectra for BaNbO3 considering only interband transitions and superposition of interband and intraband transitions.(C) Effect of non-stoichiometry on the average absorption spectra of degenerate gapped compounds in the Ba-Nb-O system.(D) Effect of non-stoichiometry on the average absorption spectra of degenerate gapped compounds in the Ca-Al-O system. The averaged plasma frequency over the three Cartesian directions is given for each system.View Large Image Figure ViewerDownload (PPT) The OVCs are generated by creating supercells of a base cell as OVC = N × (base) + pVm, where N and p are integer numbers. Number of electrons (Ne) in the conduction band is given per formula unit (f.u.) of the base compound (i.e., BaNbO3 or Ca6Al7O16). In general, the number of electrons in the conduction band for the degenerate gapped Ca-Al-O and Ba-Nb-O compounds can be predicted from the sum of composition-weighted formal oxidation states assuming Ba2+, Ca2+, Al3+, Nb5+, and O2− for each material. ΔECB is the occupied energy range of conduction band and Egint is the internal gap between CBM and VBM (Figure 1A) estimated from the density of states. Types of materials I, II, and III stand for non-transparent metals, potentially TCs (shown in red), and insulators, respectively. If the OVC in the lowest energy state does not have clearly defined vacancy sites, it is labeled as reconstructed. (A) Convex hull for the Ba-Nb-O system indicating the OVCs in color. In Figure 3B, the OVCs were not included in the calculations of the stability green zone of BaNbO3, but they are included in (A) and Figure 2A. This redefines the chemical potential stability zone under which the base compound can exist, and consequently the formation energy of Ba vacancy in BaNbO3 is about 0 eV under the redefined cation-poor conditions. Density of states for (B) BaNbO3, (C) Ba6Nb10O30, and (D) BaNb2O6, indicating the number of electrons in the conduction band per formula unit. (A) Convex hull for the Ca-Al-O system indicating the OVCs in color. (B) Chemical potential diagram for the Ca-Al-O system, showing the stability chemical potential zone for each stable OVC phase in colors corresponding to the ground states in (A). The gray zone corresponds to the prohibitive chemical potential stability zone of binary Ca-Al systems. In Figure 3D, the OVCs were not included in the calculations of the stability green zone of Ca6Al7O16, but they are included (A and B). This redefines the chemical potential stability zone under which the base compound can exist, and consequently the formation energy of Ca vacancy in Ca6Al7O16 is about 0 eV under the redefined cation-poor conditions. (C) Band structures for Ca6Al7O16 and its OVCs. (A) Schematic illustration of different contributions to optical properties in degenerate gapped compounds. (B) Absorption spectra for BaNbO3 considering only interband transitions and superposition of interband and intraband transitions. (C) Effect of non-stoichiometry on the average absorption spectra of degenerate gapped compounds in the Ba-Nb-O system. (D) Effect of non-stoichiometry on the average absorption spectra of degenerate gapped compounds in the Ca-Al-O system. The averaged plasma frequency over the three Cartesian directions is given for each system. Computationally, we find 25 binary and ternary ground state compounds (described in Data S1) of which Ba7Nb6O21, Ba5Nb4O15, Ba3Nb5O15, Ba7Nb8O24, Ba9Nb10O30, Ba26Nb27O81, and BaNb2O6 are OVCs (Figure 5). Here, 7:8:24, 9:10:30, and 26:27:82 phases have clearly defined vacancy sites, and 1:2:6, 3:5:15, and 5:4:15 OVCs are reconstructed compounds observed experimentally before.19Hessen B. Sunshine S.A. Siegrist T. Fiory A.T. Waszczak J.V. Structure and properties of reduced barium niobium oxide single-crystals obtained from borate fluxes.Chem. Mater. 1991; 3: 528-534Crossref Scopus (48) Google Scholar, 26Galasso F. Layden G. Ganung G. ANb2O6 and ATa2O6 phases.Mater. Res. Bull. 1968; 3: 397-407Crossref Scopus (42) Google Scholar, 27Galasso F. Katz L. Preparation and structure of Ba5Ta4O15 and related compounds.Acta Cryst. 1961; 14: 647-650Crossref Google Scholar, 31Sirotinkin S. Sirotinkin V. Trunov V. Structure of low-temperature modification of BaNb2O6.Zh. Neorgan. Khim. 1990; 35: 1609-1611Google Scholar, 32Vanderah T.A. Collins T.R. Wong-Ng W. Roth R.S. Farber L. Phase equilibria and crystal chemistry in the BaO–Al2O3–Nb2O5 and BaO–Nb2O5 systems.J. Alloys Compd. 2002; 346: 116-128Crossref Scopus (49) Google Scholar It should be noted that for the perovskite BaNbO3 (Figure 4), there has been a focused effort to quantify the occupancy of the A site via diffraction, and authors have noted that there are possible vacancies, as “…slight departures of the stoichiometry below the detection limits of the neutron powder diffraction technique cannot be excluded.”18Casais M.T. Alonso J.A. Rasines I. Hidalgo M.A. Preparation, neutron structural study and characterization of BaNbO3 - a Pauli-like metallic perovskite.Mater. Res. Bull. 1995; 30: 201-208Crossref Scopus (47) Google Scholar Indeed, several potential experimental compositions (i.e., Ba0.95NbO3 and Ba0.97NbO3) nearly match predicted phases (Ba26Nb27O81 ≈ Ba0.963NbO3), whereas other predicted compositions (i.e., Ba7Nb8O24 and Ba9Nb10O30) exceed the observed vacancy concentration. We identify computationally a total of ten binary and ternary ground state compounds (described in Data S2) of which Ca11Al14O32 and Ca23Al28O64 are OVCs (Figure 5). Here, both OVCs have clearly defined vacancy sites with the long-range ordering of Ca vacancies. Despite the limited information on the formation of non-stoichiometric Ca6−xAl7O16 systems (Figure 4), Ca11.3Al14O32.3 compound with Ca6Al7O16-like structure and partial occupancy of Ca sites has been reported experimentally,30Christensen A.N. Neutron powder diffraction profile refinement studies on Ca11.3Al14O32.3 and CaCiO(D0.88H0.12).Acta Chem. Scand. A. 1987; 41: 110-112Crossref Google Scholar which indirectly support our theoretical predictions. We find computationally a total of four binary and ternary ground state compounds (described in Data S3): Al2O3, Ag2O, AgAl, and AgAlO2. None of them are OVCs. In fact, the Ag3Al22O34 and its Ag2Al22O34 OVC lie energetically above the convex hull by 0.034 and 0.01 eV/atom, respectively. Specifically, Ag3Al22O34 decomposes to AlAgO2, Ag, and Al2O3 phases, while Ag2Al22O34 decomposes to AgAlO2 and Al2O3. In other words, the formation of Ag vacancy is highly effective and drains all electrons from the CB; thus, Ag2Al22O34 is an insulator. This vacancy formation increases the system stability (the energy above the convex hull is smaller for Ag2Al22O34 than that for Ag3Al22O34), but both Ag3Al22O34 and Ag2Al22O34 are unstable with respect to competing phases. Our experimental attempts to reduce synthesized Ag3Al22O34.5 to Ag3Al22O34 via heating the compound to 700°C to liberate oxygen results in the formation of free metallic silver and Ag2.5Al22O34.25 (Note S1). These results suggest that despite the fact that Ag3Al22O34 might be attractive as an intrinsic TC5Zhang X. Zhang L. Perkins J.D. Zunger A. Intrinsic transparent conductors without doping.Phys. Rev. Lett. 2015; 115: 176602Crossref PubMed Scopus (31) Google Scholar if Ag vacancy formation can be partially inhibited, the system is not likely to be realized experimentally under normal conditions. For the Ba-Nb-O system (Figure 5), among the ground state structures, the phases 1:1:3, 7:6:21, 26:27:81, 9:10:30, 7:8:24, and 3:5:15 have electrons in the CB and wide internal band gaps. From the sequences of OVCs, we find that both Ba and Nb vacancies act as acceptors, removing 2e and 5e per vacancy from the CB (Figures 6B–6D). The electronic pr