A new modeling framework for piezoresponse force microscopy

Piezoresponse force microscopy (PFM) has evolved into useful tool for measurement of local functionality of ferroelectric materials which shows great potential for applications such as data storage, ferroelectric lithography and nonvolatile memories. Better understanding of current techniques which are applied in the scale of single grain requires a straightforward analytical theory to map the PFM response for a wide range of typical experimental parameters. To this end, a new modeling framework is presented for a PFM which is modeled as a suspended cantilever beam with a tip mass. More specifically, the beam is considered to vibrate in all three directions, while subjected to a bias input voltage. The Hamilton's principle is used to derive the governing equations. The local electrostatic forces on the tip and distributed forces acting on the cantilever are also taken into account in the current modeling framework. Since the sample and tip are in the contact mode and any changes in the topography of surface will affect the indentation depth of indenter, the boundary control input force is used at the base unit. Moreover, the free end of beam with the equivalent mass of tip is connected to springs in the vertical, longitudinal and lateral directions to represent the resistance of piezoelectric material to tip movement. It is shown that the vertical bending is coupled to longitudinal displacement and lateral bending is coupled to torsion through the friction between tip and sample.