Ferroelectric phase transition and crystal asymmetry monitoring of $SrTiO_3$ using quasi $TE_{m,1,1}$ and quasi $TM_{m,1,1}$ modes
Dielectric spectroscopy of $SrTiO_3$ single crystal over a broad range of microwave frequency using quasi $TE_{m,1,1}$ and quasi $TM_{m,1,1}$ modes reveals crystal asymmetry from typical measurement of $Q$-factor, transmission or frequency characteristics in continuous cooling down to a few Kelvin. The properties of the modes due to the crystal asymmetry is validated by implementing a quasiharmonic phonon approximation. The observed ferroelectric phase transition temperature is around $51~K$, and quantum-mechanical stabilization of the paraelectric phase arises below $5~K$ with very high permittivity. Also, an antiferroelectric distortive transition was indicated at $105~K$. Landau's theory of correlation length supports the observation of an extra loss term so the transition may be identified near the $Q$-factor maxima or transmission maxima, depending on the other loss terms present in the cavity. Thus, the ferroelectric phase transition with respect to temperature may be identified when this extra-loss term causes a discontinuity in the derivative of the temperature characteristic near the minimum of total cavity loss (maxim Q-factor or maximum transmission temperature characteristic). This temperature is confirmed by transmission amplitude variation under 200 V dc electric field showing existence of the soft-mode. These measurements support a typical polarization model and explicit temperature dependency of the soft-mode incorporating an imaginary frequency.