Multivariate temperature-series analysis of stress-induced ferroelectricity in SrTiO₃: a machine learning approach with K-shape clustering and hierarchical Bayesian estimation

ABSTRACT

A new machine learning approach that transforms time-series analysis into temperature-series analysis is introduced to analyze stress-induced ferroelectricity in SrTiO₃ at 231 MPa using birefringence images observed at successive temperatures. The spatial distribution of the temperature-series data for each of the 42,280 pixels was clustered using the multivariate $\mathit{K}$-shape clustering method based on the shape similarity of the temperature dependence. In addition, to obtain the structural and ferroelectric phase transition temperatures, $T_{\mathrm{c}}$ and $T_{\mathrm{F}}$, hierarchical Bayesian temperature-series estimation was performed at each pixel (as a lower level) constrained over the entire cluster (as a higher level) considering the measurement error. Consequently, the K-shape clustering method revealed four clusters corresponding to elongated ferroelectric domains, explained by slight differences in retardance and fast-axis direction. Statistical analysis of the Bayesian posterior probability distribution showed a uniform distribution of $T_{\mathrm{c}}$ over the sample, but an inhomogeneous distribution of $T_{\mathrm{F}}$. The higher $T_{\mathrm{F}}$ regions exhibited a concentration of stress and/or strain. The Pearson correlation coefficient calculations suggested a strong to moderate relationship between the distribution of T_F and the ferroelectric state, while the correlation between T_c and the ferroelectric state was weak or nonexistent. The combination of machine learning and statistics provides a more reliable and less arbitrary approach to analyzing temperature-series data. These multilevel analyses are particularly useful in studying critical phenomena near phase transition temperatures in condensed matter physics.

GRAPHICAL ABSTRACT

IMPACT STATEMENT

Temperature-series analysis, combined with statistics, provides deeper insights into numerous imaging data observed at successive temperatures. This method will bring innovation to study of critical phenomena in condensed matter physics.

KEYWORDS:

• Temperature-series analysis
• $\mathit{K}$-shape clustering
• hierarchical Bayesian estimation
• birefringence image
• stress-induced ferroelectricity
• SrTiO${ }_3$

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27660400.2024.2342234

Additional information

Funding

This research was partially supported by a Grant-in-Aid for Scientific Research (KAKENHI) Grant Numbers [JP21K04897] and [JP23K03283] from the Japan Society for the Promotion of Science.