Flexocoupling impact on the generalized susceptibility and soft phonon modes in the ordered phase of ferroics

The impact of the flexoelectric effect on the generalized susceptibility and soft phonon dispersion is not well known in the long-range-ordered phases of ferroics. Within the Landau-Ginzburg-Devonshire approach we obtained analytical expressions for the generalized susceptibility and phonon dispersion relations in the ferroelectric phase. The joint action of the static and dynamic flexoelectric effects induces nondiagonal components of the generalized susceptibility, whose amplitude is proportional to the convolution of the spontaneous polarization with the flexocoupling constants. The flexocoupling essentially broadens the k spectrum of the generalized susceptibility and leads to an additional ``pushing away'' of the optical and acoustic soft mode phonon branches. The degeneracy of the transverse optical and acoustic modes disappears in the ferroelectric phase in comparison with the paraelectric phase due to the joint action of flexoelectric coupling and ferroelectric nonlinearity. The results obtained might be mainly important for theoretical analyses of a broad spectrum of experimental data, including neutron and Brillouin scattering.