Identifying the 'inorganic gene' for high-temperature piezoelectric perovskites through statistical learning

Abstract

This paper develops a statistical learning approach to identify potentially new high-temperature ferroelectric piezoelectric perovskite compounds. Unlike most computational studies on crystal chemistry, where the starting point is some form of electronic structure calculation, we use a data-driven approach to initiate our search. This is accomplished by identifying patterns of behaviour between discrete scalar descriptors associated with crystal and electronic structure and the reported Curie temperature (T_C) of known compounds; extracting design rules that govern critical structure-property relationships; and discovering in a quantitative fashion the exact role of these materials descriptors. Our approach applies linear manifold methods for data dimensionality reduction to discover the dominant descriptors governing structure-property correlations (the 'genes') and Shannon entropy metrics coupled to recursive partitioning methods to quantitatively assess the specific combination of descriptors that govern the link between crystal chemistry and T_C (their 'sequencing'). We use this information to develop predictive models that can suggest new structure/chemistries and/or properties. In this manner, BiTmO₃-PbTiO₃ and BiLuO₃-PbTiO₃ are predicted to have a T_C of 730^°C and 705^°C, respectively. A quantitative structure-property relationship model similar to those used in biology and drug discovery not only predicts our new chemistries but also validates published reports.

Keywords: data-driven modelling; high-temperature piezoelectrics; information theory; inorganic gene; statistical learning.

Figure 1.

In this figure, we map the Curie temperature (T_C) of known and predicted perovskite-based ferroelectric compounds in the chemical space of BiMeO₃–PbTiO₃ solid solution, where Me is a single cation with charge 3+ (e.g. Al, Sc, In, etc.) or a combination of two different cations Me_1/2Me_1/2 (e.g. ZnTi, ZnZr, ZnSn, etc.), Me_2/3Me_1/3 (e.g. ZnNb, MgNb) and Me_3/4Me_1/4 (e.g. ZnW, MgW, ScGa) with an average charge 3+ and that occupies the octahedral site of the perovskite lattice (Eitel et al. 2001; Grinberg et al. 2005; Suchomel & Davies 2005; Stein et al. 2006; Grinberg & Rappe 2007). The target design space represents the high-temperature regime that is of interest to us, and, as it can be clearly seen, the chemical search space is sparse in this region with as many as only three compounds being identified. For reference, T_C of PbZrO₃–PbTiO₃ solid solution is also indicated in this figure. Our objective is to systematically explore the complex chemical search space and identify potentially new piezoelectric materials that have high T_C. In this article, we report our computational work, where we have focused particularly on identifying a suitable Me³⁺ cation (which is weakly ferroelectrically active and occupies the octahedral site of the perovskite lattice) that can significantly enhance the T_C of BiMeO₃–PbTiO₃ solid solution. The distinction between strong and weak ferroelectric activity was made based on the degree of off-centring tendency of Me cations in MeO₆ octahedra. Filled circles, Me cations that show strong ferroelectric activity; filled squares, Me cations that show weak ferroelectric activity; filled triangles, Me cations that show strong and weak ferroelectric activity. (Online version in colour.)

Figure 2.

(a) A network of corner-sharing BO₆ octahedra with a large A-site cation occupying the interstitial position is shown. (b) The simplified unit-cell representation of cubic perovskite without showing coordination. (c) The geometry of the building units, AO₁₂ cuboctahedra and BO₆ octahedra, with 12-coordinated A-site and 6-coordinated B-site, respectively. The description of the crystal structure in the form of structural building units presents a number of diverse choices to develop new descriptors based on the site chemistry and coordination. (Online version in colour.)

Figure 3.

The univariate tolerance factor–T_C model of Eitel et al. (2001) is shown here. The shortcomings of the univariate tolerance factor–T_C model are clearly noticeable as the data show significant scatter owing to the presence of two clusters of compounds with different physics. This indicates that the tolerance factor is only a necessary condition and not sufficient for modelling T_C. We have addressed the shortcomings of the tolerance factor–T_C model by developing a multivariate model that considers six key crystal chemical descriptors instead of only the tolerance factor. Notation for chemical compounds and parameters are described in the electronic supplementary material. (Online version in colour.)

Figure 4.

Loadings plot between PC1 and PC2 showing the interactions of 30 descriptors captured by PCA. Based on the angle θ, the degree of correlation between the target variable and other attributes is established. Two zones are marked in the figure that show a strong correlation with the target variable (T_t): the red zone (with stripes) signifies attributes that show positive correlation with T_t and the green zone (no stripes) signifies variables that show negative correlation with T_t. The abbreviations of the attributes are provided in table 1. (Online version in colour.)

Figure 5.

Multivariate predicted model (abscissa) in comparison with the measured T_C as reported in the literature (Eitel et al. 2001; Grinberg et al. 2005) is shown for the PbTiO₃ end members. The model was developed by using 15 chemistries and tested for five chemistries. The new figure of merit is T_C=−(789.912×t)−(153.932×r_A)+(1013.981×r_B)+(796.5864×d_B–O)−(138.9× d_A–O)−(55.6076×B_EN)−526.537. Based on the new figure of merit, the T_C of new piezoelectric chemistries BiTmO₃–PT and BiLuO₃–PT were predicted to be 730^°C and 705^°C, respectively (labelled red in the figure). It should be noted that the T_C of BiTmO₃–PT and BiLuO₃–PT plotted in the figure is only the predicted value and needs to be experimentally validated. Notation for chemical compounds and parameters are described in the electronic supplementary material. Filled circles, training set; filled triangles, test set; plus symbols, new predictions. (Online version in colour.)

Figure 6.

The dendrogram (or tree diagram) classification model developed based on the recursive partitioning method for identifying new potentially stable perovskite compounds is shown. We used the Shannon entropy as a selection criterion to identify key descriptors, and a hierarchical set of design rules were formulated to develop classification schemes that have been approached by empirical observation. The leaf nodes that are labelled ‘yes’ or ‘no’ indicate compounds that may have a stable perovskite structure-type or not a perovskite, respectively. From the dendrogram, 11 design rules were formulated for testing the perovskite structural stability. By applying the dendrogram to the four candidate high-temperature materials BiErO₃, BiHoO₃, BiTmO₃ and BiLuO₃, only two compounds, BiTmO₃ and BiLuO₃, were identified as having the stable perovskite crystal structure at high-pressure/-temperature conditions. As a result, BiTmO₃–PbTiO₃ and BiLuO₃–PbTiO₃ solid solutions were identified as new perovskite compounds with a significantly high T_C while having piezoelectric behaviour. The dendrogram application of other Bi-based systems BiMEO₃, where ME=Cr, Co, Ga and Ni, also identifies them as having the perovskite crystal structure in agreement with the literature (Ishiwata et al. 2002; Baettig et al. 2005; Goujon et al. 2008; Oka et al. 2010). In the dendrogram, d_A–O is the ideal A–O bond length calculated based on the bond-valence method, t_IR is the tolerance factor from ionic radii data, r_A is ionic radii (Shannon’s scale) of the A-site cation with coordination number 12, r_B is the ionic radii (Shannon’s scale) of the B-site cation with coordination number 6, B_EN–O_EN is the electronegativity difference (Pauling’s scale) between B-site and O-site, A-ionicity is the product of r_A/r_O and A_EN–O_EN, B-ionicity is the product of r_B/r_O and B_EN–O_EN and GII is the global stability index (Zhang et al. 2007). (Online version in colour.)