Flexoelectricity

Flexoelectricity is a property of a dielectric material where there is coupling between electrical polarization and a strain gradient. Flexoelectricity is closely related to piezoelectricity, but while piezoelectricity refers to polarization due to uniform strain, flexoelectricity refers specifically to polarization due to strain that changes from point to point in the material. This nonuniform strain breaks centrosymmetry, meaning that unlike in piezoelectricity, flexoelectric effects occur in both centrosymmetric and asymmetric crystal structures. Flexoelectricity is not the same as Ferroelasticity.

In common useage flexoelectricity is the generation of polarization due to a strain gradient; inverse flexoectricity is when polarization, often due to an applied electric field, generates a strain gradient. Converse flexoelectricity is where a polarization gradient induces strain in a material.

The electric polarization $$P_i$$ due to mechanical strain of $$\epsilon_{ij}$$ in a dielectric is given by
 * $$P_i=e_{ijk}\epsilon_{jk}+\mu_{ijkl}\frac{\partial\epsilon_{jk}}{\partial x_l}$$

where the first term corresponds to the direct piezoelectric effect and the second term corresponds to the flexoelectric polarization induced by the strain gradient.

Here, the flexoelectric coefficient, $$\mu_{ijkl}$$, is a fourth-rank polar tensor and $$e_{ijk}$$ is the coefficient corresponding to the direct piezoelectric effect.