Insights into Cation Ordering of Double Perovskite Oxides from Machine Learning and Causal Relations

This work investigates origins of cation ordering in double perovskites using first-principles theory computations combined with machine learning (ML) and causal relations. We have considered various oxidation states of A, A′, B, and B′ from the family of transition metal ions to construct a diverse compositional space. A conventional framework employing traditional ML classification algorithms such as Random Forest (RF) coupled with appropriate features including geometry-driven and key structural modes leads to accurate prediction (∼98%) of A-site cation ordering. We have evaluated the accuracy of ML models by employing analyses of decision paths, assignments of probabilistic confidence bound, and finally a direct non-Gaussian acyclic structural equation model to investigate causality. Our study suggests that structural modes are crucial for classifying layered, columnar, and rock-salt ordering. The charge difference between A and A′ is the most important feature for predicting clear layered ordering, which in turn depends on the B and B′ charge separation. We have also designed mathematical relationships with these features to derive energy differences to form clear layered ordering. The trilinear coupling between tilt, in-phase rotation, and A-site antiferroelectric displacement in the Landau free-energy expansion becomes the necessary condition behind formation of A-site cation ordering.