Effect of flexoelectricity on band structures of one-dimensional phononic crystals

As a size-dependent theory, flexoelectric effect is expected to be prominent at the small scale. In this paper, the band gap structure of elastic wave propagating in a periodically layered nanostructure is calculated by transfer matrix method when the effect of flexoelectricity is taken into account. Detailed calculations are performed for a BaTiO₃-SrTiO ₃ two-layered periodic structure. It is shown that the effect of flexoelectricity can considerably flatten the dispersion curves, reduce the group velocities of the system, and decrease the midfrequency of the band gap. For periodic two-layered structures whose sublayers are of the same thickness, the width of the band gap can be decreased due to flexoelectric effect. It is also unveiled from our analysis that when the filling fraction is small, wider gaps at lower frequencies will be acquired compared with the results without considering flexoelectric effect. In addition, the band gap structures will approach the classical result as the total thickness of the unit cell increases. Our results indicate that the scaling law does not hold when the sizes of the periodic structures reach the nanoscale dimension. Therefore, the consideration of flexoelectric effect on the band structure of a nanosized periodic system is significant for precise manipulation of elastic wave propagation and its practical application.