Surface and volume modes of polarization waves in ferroelectric films

We investigate the characteristic modes of polarization waves in ferroelectric films. This is motivated by the recent surge of interest in the excitations of the ferroelectric order inspired by the duality between electric dipoles in ferroelectrics and magnetic dipoles in ferromagnets that has led to the introduction of the area of ferronics by analogy to magnonics. We report that a ferroelectric film supports surface and volume modes of polarization waves, analogous to the surface and volume magnetostatic spin-wave modes in a ferromagnetic film. However, while in ferromagnets each type of mode has only one (positive) frequency band, in a ferroelectric film the surface and volume modes have two frequency bands each. We present the dependence of the frequencies on the wave vector for both modes for the parameters of the classic ferroelectric $\mathrm{LiNb}{\mathrm{O}}_{3}$, with polarization either parallel or perpendicular to the film. The frequencies lie in the low terahertz band that is experimentally accessible.