  Electromechanical hysteresis loop formation in piezoresponse force microscopy
of thin ferroelectric films is studied with special emphasis on the effects of
tip size and film thickness, as well as dependence on the tip voltage
frequency. Here, we use a combination of Landau-Ginzburg-Devonshire (LGD)
theory for the description of the local polarization reversal with decoupling
approximation for the calculation of the local piezoresponse loops shape,
coercive voltages and amplitude. LGD approach enables addressing both
thermodynamics and kinetics of hysteresis loop formation. In contrast to the
"rigid" ferroelectric approximation, this approach allows for the piezoelectric
tensor components dependence on the ferroelectric polarization and dielectric
permittivity. This model rationalizes the non-classical shape of the dynamic
piezoelectric force microscopy (PFM) loops.
