The electrode size dependence of the effective large signal piezoelectric response coefficient (d 33,f ) of lead zirconate titanate (PZT) thin films is investigated by using double beam laser interferometer measurements and finite element modeling. The experimentally observed electrode size dependence is shown to arise from a contribution from the substrate. The intrinsic PZT contribution to d 33,f is independent of electrode size and is equal to the theoretical value derived assuming a rigid

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... bstrate. The substrate contribution is strongly dependent on the relative size of the electrode with respect to the substrate thickness. For electrode sizes larger than the substrate thickness, the substrate contribution is positive and for electrode sizes smaller than the substrate thickness, the substrate contribution is negative. In the case of silicon substrates, if the electrode size is equal to the substrate thickness, the substrate contribution vanishes, and the measured value of d 33,f is equal to the theoretical value under the rigid substrate assumption. V C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4821948] Piezoelectric and ferroelectric thin films are used in a number of electro-mechanical systems including sensors and actuators. Accurate knowledge of the piezoelectric material properties is crucial for a successful design of such systems. Double Beam Laser Interferometry (DBLI) is a proven, powerful technique for the measurement of the out-of-plane piezoelectric coefficient. 1-3 Nevertheless, since the film is clamped by the substrate, the measured effective piezoelectric coefficient (d 33,f ) is related to both the out-of-plane (d 33 ) and in-plane (d 31 ) piezoelectric coefficients in addition to the elastic properties of the film and the substrate. A simple theoretical model for d 33,f was developed by Lefki and Dormans by assuming completely rigid clamping of the film by the substrate. 4 Considering the elastic nature of the substrate, the clamping by the substrate is never completely rigid. The influence of the elastic properties of the substrate on the value of d 33,f has been studied by finite element model simulations. 5 Moreover, the measured d 33,f has been shown to have a strong dependence on the top electrode size. 6 Even though it has been recognized that the elastic nature of the substrate plays a role in the electrode size effect, 7 the exact mechanism of this influence has not been well understood. It is generally believed that the mechanics of the layered structure of the film and the substrate is responsible for the observed electrode size effect on d 33,f . It is also commonly assumed that the saturated value of d 33,f vs. electrode size is equal to the true value of d 33,f . In this paper, we present a detailed investigation of the mechanism of interaction between the substrate and the piezoelectric thin film that is responsible for the electrode size effect using measurements and finite element modeling. Our study clearly shows that the electrode size dependence of d 33,f is primarily due to the substrate contribution. The stresses induced in the substrate due to the mechanical interaction between the piezoelectric film and the substrate cause a small thickness change in the substrate that adds to the expansion of the piezoelectric film when an electric field is applied across the film. Furthermore, we show that this effect is a strong function of the relative size of the electrode with respect to the substrate thickness. In fact, the theoretical value of d 33,f defined under the rigid substrate assumption is obtained when the electrode size is approximately equal to the substrate thickness for silicon substrates. The samples used in this study are lead zirconate titanate (PZT) thin films deposited using a sol-gel process on 8 00 Si substrates and are approximately 1.9 lm thick. The bottom electrode is 100 nm thick platinum and the top electrode is 100 nm thick ruthenium. Top electrodes were defined by photolithography and consisted of both square and circular contacts with sizes ranging from 0.1 mm to 2 mm. An aixACCT DBLI system (model: aixDBLI industrial line) was used to measure small and large signal d 33,f at different field strengths, although only large signal measurement results are presented here. Hence, in this work, we label our measurement results as d 33,f,ls to distinguish them from the simulation results which are labeled as d 33,f . The samples exhibit extremely uniform piezoelectric properties across the 8 00 wafer with an average measured large signal (d 33,f,ls ) value of 185 pm/V and a standard deviation of 4.5 pm/V at 150 kV/cm for 2 mm 2 mm square electrodes (based on measurements at 43 different locations across the wafer). a) Present address: