Landau-Ginzburg-Devonshire theory for electromechanical hysteresis loop formation in piezoresponse force microscopy of thin films

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.