Accelerated search for BaTiO3-based piezoelectrics with vertical morphotropic phase boundary using Bayesian learning

Abstract

An outstanding challenge in the nascent field of materials informatics is to incorporate materials knowledge in a robust Bayesian approach to guide the discovery of new materials. Utilizing inputs from known phase diagrams, features or material descriptors that are known to affect the ferroelectric response, and Landau-Devonshire theory, we demonstrate our approach for BaTiO₃-based piezoelectrics with the desired target of a vertical morphotropic phase boundary. We predict, synthesize, and characterize a solid solution, (Ba_0.5Ca_0.5)TiO₃-Ba(Ti_0.7Zr_0.3)O₃, with piezoelectric properties that show better temperature reliability than other BaTiO₃-based piezoelectrics in our initial training data.

Keywords: Bayesian learning; Pb-free materials; materials informatics; morphotropic phase boundary; piezoelectric materials.

Fig. 1.

Importance of vertical MPB for superior piezoelectric performance in the composition–temperature phase diagram for ferroelectric systems. A vertical MPB, as schematically shown in A, provides temperature-independent $d_{33}$ piezoelectric property (Right) for the MPB composition (X_MPB), which is desirable for practical applications. In contrast, a tilted MPB (as shown in B) gives rise to highly temperature-sensitive $d_{33}$ property (Right), which is undesirable. Notice that in A with the vertical MPB, $d_{33}$ max is sustained for a larger temperature interval for the X_MPB composition. However, in B for the composition X_MPB, $d_{33}$ max occur only at room temperature (RT), because only at this temperature X_MPB resides at the MPB. All known BaTiO₃-based piezoelectric materials exhibit tilted MPB as shown in B. Our objective is to discover new chemical compositions that may have the desired vertical MPB in BaTiO₃-based solid solutions.

Fig. 2.

Bayesian learning for materials design. An initial experimental dataset of 19 phase diagrams and features that are known to affect the MPB serve as input to learning. The Landau functional form for MPB serves as prior knowledge to constraint the model space. The resulting Bayesian linear regression model relates phase boundaries (${\tau }_{MPB}$ and ${\tau }_{PF}$) to materials features, where ${\tau }_{MPB}$ and ${\tau }_{PF}$ are MPB and paraelectric-to-ferroelectric phase boundaries, respectively. The model is trained and cross-validated with the initial data. The trained model with updated prior and posterior distributions is applied to a dataset of unexplored phase diagrams to predict $dx$. The “best” candidate with smallest $dx$ is chosen for synthesis and characterization. We then augment the initial dataset with the newly measured $dx$ to update the inference model. The loop is repeated until the material with the desired response is discovered.

Fig. 3.

(A) Predicted (solid lines) vs. experimental (dots) phase diagram for BZT-$m$50-$n$30 from Bayesian linear regression. The blue and red solid lines show the mean phase boundaries, and the blue and red dashed lines mark the 95% confidence intervals. (B) Updated Bayesian linear regression model after augmenting the experimental BZT-$m$50-$n$30 data. Notice the reduction in uncertainties (dashed lines) for the updated model.

Fig. 4.

Results from experiment. (A) Phase diagram of newly found BZT-$m$50-$n$30 system, comparing with that of best in the training set. (B) The temperature dependence of normalized $d_{33}$ for the newly found BZT-$m$50-$n$30 system, compared with the best in the training set. (Inset) Measured $d_{33}$ data as a function of temperature for BZT-$m$50-$n$30 and best in the training set.