Critical electrode size in measurement ofd₃₃coefficient of films via spatial distribution of piezoelectric displacement

Spatial distributions of piezoelectric displacement response across the top electrode have been used in this paper to measure the piezoelectric coefficient d33 of films based on the converse piezoelectric effect. The technical details and features of a scanning laser Doppler vibrometer have been summarized and discussed for accurately obtaining the spatial displacement distributions. Three definitions, including the apparent, the effective and the constrained piezoelectric coefficient d33 of films, have been clarified and used to better understand the fundamental phenomenon behind the measured displacement distributions. Finite element analysis reveals that both the apparent and the effective piezoelectric coefficients depend on the electrode radius of test capacitor as well as film thickness. However, there exists a critical electrode size for apparent piezoelectric coefficients and a critical test capacitor aspect ratio for effective piezoelectric coefficient. Beyond their respective critical values, both coefficients converge to the constrained piezoelectric coefficient irrespective of film thickness. The finding of the critical electric size makes it possible to consistently measure the constrained piezoelectric coefficient of films by using the spatial distributions of the piezoelectric displacement response and becomes the fundamental criterion of this measurement method.