e_31,f determination for PZT films using a conventional 'd₃₃' meter

A new and simple method is described for the determination of the piezoelectric coefficients d_33,f and e_33,f for piezoelectric films deposited on substrates using a conventional point-loading 'd₃₃' or 'Berlincourt' piezometer. An analytical mathematical model is developed which simulates the dynamical flexure of such films when a ring-supported sample is subject to central loading. Classical plate theory and elastic analysis are used to calculate the stresses in doped lead zirconate titanate (PZT) film for different radii of supporting rings, enabling both piezoelectric coefficients to be determined through a simple modification to the piezometer. The analytical model for the radial stresses has been evaluated in comparison with the results of finite element analysis and has shown a good correlation. The new measurement technique has been applied to both thick films of PZT and thin films of manganese-doped lead zirconate titanate (PMZT) on silicon substrates. The values of d_33,f and e_31,f obtained experimentally are found to be similar to those that have been determined by more elaborate methods.