A phonon fluctuation state in the stuffed tridymite-type oxides Ba$_{1-x}$Sr$_x$Al$_2$O$_4$

The paraelectric phase (space group $P6_322$) of the stuffed tridymite-type ferroelectric BaAl$_2$O$_4$ has recently been reported to possess the energetically competing low-energy phonon modes at the M- and K-points, both of which soften at $T_{\rm C}$ simultaneously [Y. Ishii et al., Phys. Rev. B 93, 134108 (2016).]. The M-point mode condenses at $T_{\rm C}$=450 K, giving rise to the low-temperature ferroelectric phase ( $P6_3$), whereas the K-point mode is electrostatically unfavorable and just disappears below $T_{\rm C}$. In this study, we investigated the thermal diffuse scatterings in the electron diffraction and the dielectric properties of Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ ($x = 0-0.5$). We present that Ba$_{1-x}$Sr$_x$Al$_2$O$_4$ system exhibits a "fluctuating'' state, in which the M-point soft mode does not condense but survives and fluctuates down to low temperature, below $T^*\sim200$ K. Although the K-point soft mode disappears below $T^*$, the $P6_322$ crystal structure is retained at temperatures down to 15 K. The wave vector of the M-point diffuse scatterings is temperature dependent below $T^*$ and loses commensurateness as the temperature decreases. This result indicates that the fluctuation in the wave vector of the M-point mode increases at low temperatures.