  The evolution of the atomic structures of the combinatorial library of
Sm-substituted thin film BiFeO3 along the phase transition boundary from the
ferroelectric rhombohedral phase to the non-ferroelectric orthorhombic phase is
explored using scanning transmission electron microscopy (STEM). Localized
properties including polarization, lattice parameter, and chemical composition
are parameterized from atomic-scale imaging and their causal relationships are
reconstructed using a linear non-Gaussian acyclic model (LiNGAM). This approach
is further extended toward exploring the spatial variability of the causal
coupling using the sliding window transform method, which revealed that new
causal relationships emerged both at the expected locations, such as domain
walls and interfaces, but also at additional regions forming clusters in the
vicinity of the walls or spatially distributed features. While the exact
physical origins of these relationships are unclear, they likely represent
nanophase separated regions in the morphotropic phase boundaries. Overall, we
pose that an in-depth understanding of complex disordered materials away from
thermodynamic equilibrium necessitates understanding not only of the generative
processes that can lead to observed microscopic states, but also the causal
links between multiple interacting subsystems.
