The piezoelectric response of a material under a nanoscale biased tip scanned across a sample in piezoelectric force microscopy (PFM) provides insight into the structure and dynamics of domain walls in ferroelectrics. While the vertical displacements of the tip under piezoelectric deformations of the sample have been reasonably explained, the origin of the lateral twisting of the tip remains unclear. This poses a serious problem when combining vertical and lateral signals to create vector PFM maps of polarization distribution in ferroelectrics. Using a combination of finite element modeling and analytical theory, and by comparison with prior experimental work across a single antiparallel domain wall on the (0001) surface of LiNbO${}_3$, we unequivocally show that the lateral signal originates from a shear displacement of the surface. We show that there are two types of lateral signals, one arising from the $d_{15}$ shear deformation, and the other from the $d_{22}$ lateral deformation. The vertical PFM signal surprisingly shows equal contributions from the $d_{33}$ (leading to normal displacements) and $d_{15}$ (leading to shear displacement) coefficients. We also show that an averaging of the PFM signal over a finite contact area of the tip, as experimentally observed, is essential to understanding the line shape of the PFM responses across the wall. After clarifying the origin of the nanoscale PFM signals, we conclude that, in general, a vertical signal does not automatically indicate a polarization component out of the surface, while a lateral signal does not automatically indicate an in-plane polarization component. Without a detailed theory or simulation especially in materials with nanoscale domain structures, ferroelectric relaxors, and morphotropic compositions, such assumptions may lead to incorrect domain and wall interpretations. The proposed model and numerical simulation method could be applied to all piezoelectric materials.

• Received 26 May 2012

DOI:https://doi.org/10.1103/PhysRevB.86.134115