Stoney formula for piezoelectric film/elastic substrate system

With the trends in miniaturization, and particularly the introduction of micro- and nano-electro-mechanical system, piezoelectric materials used in microelectronic devices are deposited usually in the form of thin film on elastic substrates. In this work, the bending of a bilayer comprising a piezoelectric film deposited on an elastic substrate, due to the mismatch, is investigated. An analytic formula relating the curvature of the bilayer to the mismatch, the electroelastic constants and the film thickness is obtained, and from this formula, a transverse piezoelectric constant d₃₁ can be estimated. Meanwhile the influence of electromechanical coupling coefficient on the curvature is discussed.