Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect

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---the coupling of strain gradient to polarization---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 $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$.