System and method for extraction of piezoelectric constants electrically

A method for determining piezoelectric parameters of a piezoelectric structure comprising:

• applying a voltage to thereby applying a biasing electrical field E over the piezoelectric structure, where the applied voltage is adapted so that the biasing electrical field has one or more selected directions in relation to a poling direction P of said piezoelectric structure including at least one of the same and the opposite direction to the poling direction P to force the dipoles to align in a poling direction;
• determining the longitudinal piezoelectric voltage constant d33 and transversal piezoelectric voltage constant d31 of said piezoelectric structure in relation to a predetermined parameter ratio solely by knowing the applied electrical field E and the thickness of the piezoelectric layers without requiring any knowledge and information about the change in dielectric constant, wherein the piezoelectric structure is a thin film piezoelectric structure including a first piezoelectric layer having a first thickness and constituting a first outer electrode and a second piezoelectric layer having a second thickness and constituting a second outer electrode, and wherein determining of the piezoelectric parameters of the piezoelectric structure is further comprising:
• incorporating the piezoelectric structure in a resonator structure having a measurable resonance frequency f and an effective inductor L;
• driving the resonator against and along polarization/poling directions;
• measuring the resonance frequency f for a selection of values for the applied electric field E and wherein the predetermined parameter ratio is a capacitance ratio Cr=C↓↑/C↑↑ based on the quota between a first capacitance C↓↑ of the piezoelectric structure due to an applied biasing electrical field with a direction antiparallel to said poling direction and a second capacitance C↑↑ of the piezoelectric structure due to an applied biasing electrical field with a direction parallel with said poling direction;
• and
• determining the piezoelectric voltage constants d33 and d31 based on the relation with a normalized frequency ratio frn as a function of applied voltage, wherein the applied voltage is expressed in terms of electric field E, according to the following equation (24)
• frn=1+(2d31+d33)E+(2d31d33+d312)E2+d312d33E3
• and by fitting values for the normalized frequency ratio frn versus values of the applied electric field E using a fitting method, wherein the fitting method is a cubic equation fitting or a quadratic equation fitting.