Probing phonon softening in ferroelectrics by scanning probe microwave spectroscopy

Microwave measurements have recently been successfully applied to measure ferroelectric materials on the nanoscale, including detection of polarization switching and ferroelectric domain walls. Here we discuss the question of whether scanning probe microscopy operating at microwave frequency can identify the changes associated with the soft phonon dynamics in a ferroic. The analytical expressions for the electric potential, complex impedance, and dielectric losses are derived and analyzed, since these physical quantities are linked to experimentally measurable properties of the ferroic. As a ferroic we consider virtual or proper ferroelectric with an optic phonon mode that softens at a Curie point. We also consider a decay mechanism linked to the conductance of the ferroic, thus manifesting itself as the dielectric loss in the material. Our key finding is that the influence of the soft phonon dispersion on the surface potential distribution, complex impedance, and dielectric losses are evidently strong in the vicinity (similar to 10-30 K) of the Curie temperature. Furthermore, we quantified how the spatial distribution and frequency spectra of the complex impedance and the dielectric losses react on the dynamics of the soft phonons near the Curie point. These results set the stage for characterization of polar phase transitions with nanoscale microwave measurements, providing a complementary approach to well established electromechanical measurements for fundamental understanding of ferroelectric properties as well as their applications in telecommunication and computing.