Modelling of the vertical deflection of ferroelectric bending tongues loaded at their free end

In this work, a model to describe the vertical deflection of ferroelectric bending tongues with a load at their free end is proposed. The model is based on the ferroelectric switching criterion developed by Hwang et al. ["Ferroelectric/ferroelastic interactions and a polarization switching model," Acta Metall. Mater. 43, 2073-2084 (1995)] and the Euler-Bernoulli beam theory. It notably takes into account the geometry of the bending tongue, the mechanical and piezoelectric material properties, the applied electrical field, the crystallographic state of the ferroelectric thin film and its built-in strain. Hwang's model is improved by incorporating strain saturation at high field, as expected for the butterfly loop. This allows accurate estimates of the vertical deflection for ferroelectric bending tongue based applications.