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Modeling functional piezoelectricity in perovskite superlattices with competing instabilities

Based on the locality principle of insulating superlattices, we apply the method of Wu {\it et al} [Phys. Rev. Letter {\bf 101}, 087610 (2008)] to the piezoelectric strains of individual layers under fixed displacement field. For a superlattice of arbitrary stacking sequence an accurate model is acquired for predicting piezoelectricity. By applying the model in the superlattices where ferroelectric and antiferrodistortive modes are in competition, functional piezoelectricity can be achieved. A strong nonlinear effect is observed and can be further engineered in the PbTiO$_3$/SrTiO$_3$ superlattice and an interface enhancement of piezoelectricity is found in the BaTiO$_3$/CaTiO$_3$ superlattice.