Vibration analysis of vector piezoresponse force microscopy with coupled flexural-longitudinal and lateral-torsional motions

Piezoresponse force microscopy (PFM) has evolved into a useful tool for measurement of local properties of piezoelectric materials with great potential in applications such as data storage, ferroelectric lithography and nonvolatile memories. In order to utilize PFM for low dimensional materials characterization, a comprehensive analytical modeling based on the coupled motion of PFM in all three directions is proposed. In this respect, the mechanical properties of sample are divided into viscoelastic and piezoelectric parts. The viscoelastic part is modeled as a spring and damper in the longitudinal, transversal and lateral directions, while the piezoelectric part is replaced with resistive forces acting at the end of microcantilever. It is shown that there is a geometrical coupling between flexural-longitudinal and lateral-torsional vibrations of microcantilever used in PFM. Moreover, assuming a general friction between tip and sample, additional coupling effect is also taken into account. Through an energy-based approach, it is seen that the PFM system can be governed by a set of coupled partial differential equations along with nonhomogeneous and coupled boundary conditions. A general formulation is then derived for the mode shape, frequency response, and state-space representation of system. Numerical simulations indicate that mode shapes, natural frequencies and time responses of microcantilever beam are heavily dependent on the viscoelastic and piezoelectric properties of the samples. Moreover, the results demonstrate that utilizing only transversal vibration is not a valid strategy for quantifying mechanical properties of materials with arbitrary crystallographic orientation. Hence, the proposed model with the built-in coupling effects can be a key development for acquiring precise measurements.