Flexoelectricity
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Flexoelectricity is a property of a dielectric material whereby it exhibits a spontaneous electrical polarization induced by 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 piezoelectiricty, flexoelectric effects can occur in centrosymmetric crystal structures.<ref>Pavlo Zubko, Gustau Catalan, and Alexander K. Tagantsev (2013). "Flexoelectric Effect in Solids". Annual Review of Materials Research. 43: 387–421. Bibcode:2013AnRMS..43..387Z. doi:10.1146/annurev-matsci-071312-121634. hdl:10261/99362.{{cite journal}}: CS1 maint: multiple names: authors list (link)</ref> Flexoelectricity is not the same as Ferroelasticity.
Inverse flexoelectricity, quite intuitively can be defined as generation of strain gradient due to polarization. Similarly extending on that, Converse flexoelectricity would refer to the process where a polarization gradient induces strain in a material.<ref>Abdollahi A, Domingo N, Arias I, Catalan G (2019). "Converse flexoelectricity yields large piezoresponse force microscopy signals in non-piezoelectric materials". Nature Communications. 10 (1): 1266. Bibcode:2019NatCo..10.1266A. doi:10.1038/s41467-019-09266-y. PMC 6427004. PMID 30894544.</ref>
The electric polarization <math>P_i</math> due to mechanical strain of <math>\epsilon_{ij}</math> in a dielectric is given by
- <math>P_i=e_{ijk}\epsilon_{jk}+\mu_{ijkl}\frac{\partial\epsilon_{jk}}{\partial x_l}</math>
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, <math>\mu_{ijkl}</math>, is a fourth-rank polar tensor and <math>e_{ijk}</math> is the coefficient corresponding to the direct piezoelectric effect.
