Flexoelectricity

Flexoelectricity is the linear coupling between electric polarization and a gradient of mechanical strain in dielectric materials. In tensor form, the polarization P_i is proportional to the gradient of the strain ε_jk: P_i = μ_ijkl ∂ε_jk/∂x_l, where μ_ijkl is the flexoelectric coefficient tensor. The strain gradient acts as the driving field for polarization, so a nonuniform deformation is required to generate an electric response.

Unlike piezoelectricity, which requires non-centrosymmetric crystals, flexoelectricity is symmetry-allowed in all dielectric crystals, including centrosymmetric ones. The effect is typically small in bulk materials with uniform strain but becomes significant when large strain gradients exist, such as at surfaces, interfaces, or in nanoscale structures. The magnitude and anisotropy depend on the material and geometry; in practice, flexoelectric coefficients span broad ranges and can be enhanced in ferroelectrics and perovskites.

Materials and devices that exhibit flexoelectric responses include ceramics, polymers, and oxide thin films. In nanoscale systems, bending, peeling, or curvature can produce substantial strain gradients, enabling measurable polarization without ferroelectric ordering. This has spurred research into flexoelectric contributions to size-dependent ferroelectricity, domain-wall behavior, and electromechanical sensing.

Historically, the concept emerged in the 1960s from theoretical work by researchers such as Kogan and Mindlin, with the term flexoelectricity widely used in subsequent decades. Experimental evidence and quantitative measurements accumulated in the 2000s and beyond, leading to renewed interest for fundamental understanding and potential applications in micro- and nano-electromechanical systems.

See also: piezoelectricity, electrostriction, nanoscale electromechanics.