Abstract

Crystalline piezoelectric dielectrics electrically polarize upon application of uniform mechanical strain. Inhomogeneous strain, however, locally breaks inversion symmetry and can potentially polarize even nonpiezoelectric (centrosymmetric) dielectrics. Flexoelectricity\char22{}the coupling of strain gradient to polarization\char22{}is expected to show a strong size dependency due to the scaling of strain gradients with structural feature size. In this study, using a combination of atomistic and theoretical approaches, we investigate the ``effective'' size-dependent piezoelectric and elastic behavior of inhomogeneously strained nonpiezoelectric and piezoelectric nanostructures. In particular, to obtain analytical results and tease out physical insights, we analyze a paradigmatic nanoscale cantilever beam. We find that in materials that are intrinsically piezoelectric, the flexoelectricity and piezoelectricity effects do not add linearly and exhibit a nonlinear interaction. The latter leads to a strong size-dependent enhancement of the apparent piezoelectric coefficient resulting in, for example, a ``giant'' 500% enhancement over bulk properties in $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ for a beam thickness of $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Correspondingly, for nonpiezoelectric materials also, the enhancement is nontrivial (e.g., 80% for $5\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ size in paraelectric $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ phase). Flexoelectricity also modifies the apparent elastic modulus of nanostructures, exhibiting an asymptotic scaling of $1∕{h}^{2}$, where $h$ is the characteristic feature size. Our major predictions are verified by quantum mechanically derived force-field-based molecular dynamics for two phases (cubic and tetragonal) of $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$.

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