Structure and microwave dielectric characteristics of (Ba,Sr,Ca)HfO3 ceramics

In the present work, the structure and microwave dielectric properties of (Ba,Sr,Ca)HfO₃ ceramics were systematically investigated to understand the general mechanism of tuning the temperature coefficient of resonant frequency, τ_f in perovskite ceramics. Ba_1–xSr_xHfO₃ and Sr_1–yCa_yHfO₃ could form continuous solid solutions while the solid solubility of Ca in Ba_1–yCa_yHfO₃ was about 20% (in mole). τ_f changed nonlinearly with increasing tolerance factor as the result of competition between the increase in the restoring force on the ions and the increase in polarizability. Under the guidance of three microscopic mechanisms affecting τ_f, a preliminary attempt was made to explore the suitable parameters to predict the variation trend of τ_f. Normalized ionic radii and the τ_f values of the end members were selected as independent variables, and τ_f was calculated by using multiple linear regression method. For Ba_1–xSr_xHfO₃ and Sr_1–yCa_yHfO₃ with orthorhombic structures, the root mean square error between the calculated and measured τ_f was only 6.8 × 10⁻⁶ °C⁻¹. The good agreement between the calculated τ_f values and the measured ones in Ba_1–x–ySr_xCa_yHfO₃ ceramics confirmed its validity where three elements jointly occupy A-site, but it only works when there is no structural phase transition. Progresses in this research field would not only deepen our understanding of mechanism of regulating properties in multi-ion solid solutions, but also boost the developments of closely related fields such as machine learning-assisted design and high-entropy ceramics. Finally, a good combination of microwave dielectric properties was achieved in Sr_0.15Ca_0.85Hf_0.96Ti_0.040O₃: ε_r = 27.8, Qf = 36,470 GHz, τ_f = +5 × 10⁻⁶ °C⁻¹.