The influence of electric field on the anisotropic dispersion of the flexocoupling induced phonons and ferrons in van der Waals ferrielectrics

As it has been shown recently, the influence of the flexoelectric coupling (shortly “flexocoupling”) on the fluctuations of electric polarization and elastic strains can lead to the principal changes of the dispersion law of soft optical and acoustic phonons (shortly “flexophonons”) and ferrons (shortly “flexoferrons”) in van der Waals ferrielectrics. Analytical results, derived in the framework of the Landau-Ginzburg-Devonshire approach, revealed that the dispersion of flexophonons and flexoferrons is strongly anisotropic and should depend on the magnitude and direction of an applied electric field. In this work, we study the influence of the applied electric field on the anisotropic dispersion of the flexophonons and flexoferrons in van der Waals ferrielectric CuInP₂S₆. We reveal that the frequency of acoustic flexophonons and flexoferrons tends to zero at nonzero wavevectors under an increase in the applied electric field. We relate the changes to the possible appearance of a spatially modulated incommensurate polar phase induced by flexocoupling in an external field. The critical strength of flexocoupling is determined by the magnitude of the electric field and the direction of the wavevector. This allows us to propose a method for estimating the strength of flexocoupling in van der Waals ferrielectrics, provided that the frequency of the acoustic flexophonon is zeroing at the threshold value of the electric field. Since the flexoelectric coefficients are poorly known in van der Waals ferrielectrics, the obtained analytical results can be useful for their flexo-engineering.