Research ArticleMATERIALS SCIENCE

New tolerance factor to predict the stability of perovskite oxides and halides

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Science Advances  08 Feb 2019:
Vol. 5, no. 2, eaav0693
DOI: 10.1126/sciadv.aav0693
  • Fig. 1 Perovskite structure and composition.

    (A) ABX3, in the cubic single perovskite structure (Embedded Image), where the A cation is surrounded by a network of corner-sharing BX6 octahedra. (B) A2BB′X6, in the rock salt double perovskite structure (Embedded Image), where the A cations are surrounded by an alternating network of BX6 and B′X6 octahedra. In this structure, inverting the B and B′ cations results in an equivalent structure. While the ideal cubic structures are shown here, perovskites may also adopt various noncubic structures. (C) Map of the elements that occupy the A, B, and/or X sites within the 576 compounds experimentally characterized as perovskite or nonperovskite at ambient conditions and reported in (1719).

  • Fig. 2 Assessing the performance of the improved tolerance factor, τ.

    (A) A decision tree classifier determines that the optimal bounds for perovskite formability using the Goldschmidt tolerance factor (t) are 0.825 < t < 1.059, which yields a classification accuracy of 74% for 576 experimentally characterized ABX3 solids. (B) τ achieves a classification accuracy of 92% on the set of 576 ABX3 solids based on perovskite classification for τ < 4.18, with this decision boundary identified using a one-node decision tree. All classifications made by t and τ on the experimental dataset are provided in table S1. The largest value of τ in the experimental set of 576 compounds is 181.5; however, all points with τ > 13 are correctly labeled as nonperovskite and are not shown to highlight the decision boundary. The outlying compounds at τ > 10 that are labeled perovskite yet have large τ are PuVO3, AmVO3, and PuCrO3, which may indicate poorly defined radii or incorrect experimental characterization. (C) Comparison of Platt-scaled classification probabilities, P(τ), versus t. LaAlO3 and NaBeCl3 are labeled to highlight the variation in P(τ) at nearly constant t. (D) Comparison between P(τ) and the decomposition enthalpy (ΔHd) for 36 double perovskite halides calculated using density functional theory (DFT) in the Embedded Image structure in (32) and 37 single and double perovskite chalcogenides and halides in the Embedded Image structure in (33). The legend corresponds with the anion, X. Positive decomposition enthalpy (ΔHd > 0) indicates that the structure is stable with respect to decomposition into competing compounds. The green and white shaded regions correspond with agreement and disagreement between the calculated ΔHd and the classification by τ. Points of disagreement are outlined in red. CaZrO3 and CaHfO3 are labeled because they are known to be stable in the perovskite structure, although they are unstable in the cubic structure (34, 35). For this reason, the best-fit line for the chalcogenides (X = O2−, S2−, Se2−) excludes these two points.

  • Fig. 3 Map of predicted double perovskite oxides and halides.

    Lower triangle: Probability of forming a stable perovskite with the formula Cs2BB′Cl6 as predicted by τ. Upper triangle: Probability of forming a stable perovskite with the formula La2BB′O6 as predicted by τ. White spaces indicate B/B′ combinations that do not result in charge-balanced compounds with rA > rB. The colors indicate the Platt-scaled classification probabilities, P(τ), with higher P(τ) indicating a higher probability of forming a stable perovskite. B/B′ sites are restricted to ions that are labeled as B sites in the experimental set of 576 ABX3 compounds.

  • Fig. 4 The effects of ionic radii and oxidation states on the stability of single and double perovskite oxides and halides.

    Top row: X = O2− (left to right: nA = 3+, 2+, 1+). Bottom row: nA = 1+ (left to right: X = Cl, Br, I). The experimentally realized perovskites LaGaO3, Sr2FeMoO6, AgNbO3, Cs2AgInCl6, (MA)2AgBiBr6, and MAPbI3 are shown as open circles in the corresponding plot, which are all predicted to be stable by τ. The experimentally realized nonperovskites InGaO3, CoMnO3, LiBiO3, LiMgCl3, CsNiBr3, and RbPbI3 are shown as open triangles and predicted to be unstable in the perovskite structure by τ. The organic molecule, methylammonium (MA), is shown in the last two panels. While (MA)2AgBiBr6 and MAPbI3 are correctly classified with τ, only inorganic cations were used for descriptor identification; therefore, rA = 1.88 Å (Cs+) is the largest cation considered. The gray region where rB > rA is not classified because, when this occurs, A becomes B and vice versa based on our selection rule rA > rB.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/2/eaav0693/DC1

    Table S1. The 576 ABX3 used for training and testing τ.

    Table S2. Confusion matrices for τ (above) and t (below).

    Table S3. Additional information associated with Fig. 2D.

    Table S4. Double perovskite oxides and halides.

    Fig. S1. Comparing the performance of t and τ by composition.

    Fig. S2. Sigmoidal relationship between P(τ) and τ.

    Fig. S3. (t, μ) structure map for 576 ABX3 solids.

  • Supplementary Materials

    The PDF file includes:

    • Legend for table S1
    • Table S2. Confusion matrices for τ (above) and t (below).
    • Legends for tables S3 and S4
    • Fig. S1. Comparing the performance of t and τ by composition.
    • Fig. S2. Sigmoidal relationship between P(τ) and τ.
    • Fig. S3. (t, μ) structure map for 576 ABX3 solids.

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    Other Supplementary Material for this manuscript includes the following:

    • Table S1 (.csv format). The 576 ABX3 used for training and testing τ.
    • Table S3 (.csv format). Additional information associated with Fig. 2D.
    • Table S4 (.csv format). Double perovskite oxides and halides.

    Files in this Data Supplement:

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