Piezoelectric Mimicry of Flexoelectricity

The origin of “giant” flexoelectricity, orders of magnitude larger than theoretically predicted, yet frequently observed, is under intense scrutiny. There is mounting evidence correlating giant flexoelectriclike effects with parasitic piezoelectricity, but it is not clear how piezoelectricity (polarization generated by strain) manages to imitate flexoelectricity (polarization generated by strain gradient) in typical beam-bending experiments, since in a bent beam the net strain is zero. In addition piezoelectricity changes sign under space inversion but giant flexoelectricity is insensitive to space inversion, seemingly contradicting a piezoelectric origin. Here we show that, if a piezoelectric material has its piezoelectric coefficient asymmetrically distributed across the sample, it will generate a nonzero bending-induced polarization impossible to distinguish from true flexoelectricity even by inverting the sample. The effective flexoelectric coefficient caused by piezoelectricity is functionally identical to, and often larger than, intrinsic flexoelectricity: our calculations show that, for standard perovskite ferroelectrics, even a tiny gradient of piezoelectricity (1% variation of piezoelectric coefficient across 1 mm) is sufficient to yield a giant effective flexoelectric coefficient of $1\mu \mathrm{C}/\mathrm{m}$, three orders of magnitude larger than the intrinsic expectation value.

Figure 1

(a) Homogeneously poled piezoelectric beam under bending, which does not induce a nonzero net polarization because the average strain is zero. The color plot presents the electric potential distribution. (b) Piezoelectric polarization induced in a rectangular sample under tension or compression. The red arrow represents the direction of the material polarization. (c) Flexoelectric polarization induced in a cantilever beam under bending. The polarization does not change sign by reversing the beam.

Figure 2

(a) Schematic of a cantilever beam under the point load $F$. (b) Schematic of a bimorph cantilever beam where the dark layer is piezoelectric. (c) A reversed configuration of the bimorph (rotated by 180°). The white arrows represent the polarization direction of the layers. The color plot presents the electric potential ($\varphi$) distribution for each bimorph obtained from Eqs. (3) and (4), where $E_z=-{\varphi }_{,z}$. The net polarization can be obtained from the electric potential difference of the top and bottom faces, which is identical in both bimorphs.