Hybrid Improper Ferroelectricity in (Sr,Ca)₃Sn₂O₇ and Beyond: Universal Relationship between Ferroelectric Transition Temperature and Tolerance Factor in n = 2 Ruddlesden-Popper Phases

Hybrid improper ferroelectricity, which utilizes nonpolar but ubiquitous rotational/tilting distortions to create polarization, offers an attractive route to the discovery of new ferroelectric and multiferroic materials because its activity derives from geometric rather than electronic origins. Design approaches blending group theory and first principles can be utilized to explore the crystal symmetries of ferroelectric ground states, but in general, they do not make accurate predictions for some important parameters of ferroelectrics, such as Curie temperature (T_C). Here, we establish a predictive and quantitative relationship between T_C and the Goldschmidt tolerance factor, t, by employing n = 2 Ruddlesden-Popper (RP) A₃B₂O₇ as a prototypical example of hybrid improper ferroelectrics. The focus is placed on an RP system, (Sr_1-xCa_x)₃Sn₂O₇ (x = 0, 0.1, and 0.2), which allows for the investigation of the purely geometric (ionic size) effect on ferroelectric transitions, due to the absence of the second-order Jahn-Teller active (d⁰ and 6s²) cations that often lead to ferroelectric distortions through electronic mechanisms. We observe a ferroelectric-to-paraelectric transition with T_C = 410 K for Sr₃Sn₂O₇. We also find that the T_C increases linearly up to 800 K upon increasing the Ca²⁺ content, i.e., upon decreasing the value of t. Remarkably, this linear relationship is applicable to the suite of all known A₃B₂O₇ hybrid improper ferroelectrics, indicating that the T_C correlates with the simple crystal chemistry descriptor, t, based on the ionic size mismatch. This study provides a predictive guideline for estimating the T_C of a given material, which would complement the convergent group-theoretical and first-principles design approach.