High critical transition temperature of lead-based perovskite ferroelectric crystals: A machine learning study

Abstract

Ferroelectrics undergoes a reversible structural phase from the ferroelectric phase to the paraelectric phase when its temperature exceeds the critical temperature namely Curie temperature <i>T</i><sub>c</sub>. As ferro-paraelectric phase transition is always accompanied by heat-flow, dielectric and pyroelectric anomaly, the value of <i>T</i><sub>c</sub> is extremely important for ferroelectrics. In this paper, the Curie temperature of lead-based perovskite ferroelectric solid solution is studied by machine learning methods including kernel ridge regression (KRR), support vector regression (SVR) and extremely randomized trees regression (ETR). We collect the <i>T</i><sub>c</sub> values of 205 different lead-based perovskites from published experimental papers, both simple perovskites with only one type of <i>B</i> site ion and complex perovskites with up to 5 kinds of ions in <i>B</i> position such as PMN-PFN-PZT are gathered. The diversity of our dataset is guaranteed for the good generalization of our model in perovskite solid solution of different complexity. The features are constructed from the physical and chemical properties of the <i>B</i> site elements in corresponding materials. The weighted-average and variance of the elemental properties are calculated and fed to machine learning models. We use the 5 runs of ten fold cross-validation method to evaluate the machine learning models. The hyperparameters are also chosen carefully with the cross-validation to avoid over fitting. The radial basis function kernel is used in both KRR and SVR. The insensitive error in the SVR is set to be 4 which is comparable to the random error in experiment. From our cross-validation, we find that the mean average errors (MAEs) between the predicted and experimental values of the machine learning methods are 14.4 K, 14.7 K, and 16.1 K, respectively. And the root-mean-square errors (RMSEs) are 22.5 K, 23.4 K, 23.8 K, respectively. After the optimization and the evaluation, our three machine learning models are stacked together by averaging the output of each regression model and thus building an ensemble model. The MAE of the ensemble model is 13.9 K. The RMSE of the ensemble model is 21.4 K. The predicted values keep a correlation coefficient of 0.97 with the experimental values. From the variance reduction in ETR, we derive the importance of our features when determining the Curie temperatures. The five most important factors in our ETR model are " weighted-average thermal conductivity”, " weighted-average conductivity”, " variance of specific heat capacity”, " weighted-average element number”, and " weighted-average relative atomic displacement”. We predict the Curie temperatures higher than those of 200000 types of lead-based perovskites after being trained. Now, we provide two ferroelectric materials that may have high Curie temperatures: 0.02PbMn<sub>1/2</sub>Nb<sub>1/2</sub>O<sub>3</sub>-0.98PbTiO<sub>3</sub> (0.02PMN-0.98PT) and 0.02PbGa<sub>1/2</sub>Nb<sub>1/2</sub>-0.02PbMn<sub>1/2</sub>Nb<sub>1/2</sub>O<sub>3</sub>-0.96PbTiO<sub>3</sub> (0.02PGN-0.02PMN-0.96PT). The predicted Curie temperatures of them are 481 ℃ and 466 ℃, respectively.