Quantitative Piezoresponse Force Microscopy: Calibrated Experiments, Analytical Theory and Finite Element Modeling

Abstract

We present quantitative experiments, analytical theory and finite element modeling­ (FEM) of vertical and lateral piezoresponse force microscopy (PFM) across a single antiparallel (180°) ferroelectric domain wall. There are three important aspects in making quantitative measurements. (1) Calibration and background subtraction of PFM displacements; (2) characterization of the tip shape and contact area; and (3) analytical theory and numerical simulations that incorporate all the relevant property tensors (dielectric, piezoelectric, and ferroelectric), tip shape, contact geometry, and the relevant physics of the feature being studied, such as the width of the wall. By calibrating the displacement of the tip, and using a reference sample, one can measure nanoscale piezoelectric coefficients, which are shown to be independent of tip size for a uniform sample. The shape of the contact area of a tip with the sample is characterized by field emission scanning electron microscopy (FE-SEM) to be disk-like. Only a true-contact with zero dielectric gap between the tip and the sample can explain the experimental PFM wall width versus tip radius measurements. Finally, in the limit of the tip disk-radius approaching zero, one can estimate the ferroelectric wall width from the vertical PFM profiles across the wall. The most complete analytical theory and finite element modeling to date are presented that can realistically simulate the PFM profile across a single wall. While vertical PFM signal agrees well with theory and simulations, the lateral PFM signal shows excellent qualitative agreement only. The experimental width of the lateral PFM signal across a wall is significantly wider than that predicted by FEM, suggesting elements of surface physics that are not captured in the current electromechanical theory of PFM.