3D Vector Piezoresponse Imaging with Interferometric Atomic Force Microscopy

Forces acting between an Atomic Force Microscope (AFM) tip and sample are 3D. Despite this, most AFM force measurements are confined to one or two dimensions. Extending AFM force measurements into 3D has previously required complex, difficult, and time-consuming workflows. Here, an accurate, interferometric method for quantifying the full, 3D response of an AFM tip is demonstrated to localized forces. This approach is demonstrated on a series of piezoelectric materials and show that this approach yields quantitative 3D measurement independent of the sample orientation beneath the tip. This approach simplifies existing, angle-resolved piezoresponse force microscopy (PFM) techniques. These measurements benefit from the greatly reduced noise floor ( $\approx 5\mathrm{fm}/\sqrt{\mathrm{Hz}}$ ) and intrinsic accuracy of the interferometric measurements. One important result is that the vertical piezo sensitivity (d_eff,z, units of pm/V) is systematically 2-3x larger than the in-plane piezo sensitivities (d_eff,lat). A simple analysis of vertical and lateral contact stiffnesses, due to the difference in the Young (vertical) and Shear (lateral) sample moduli d_theory,z/d_theory,lat ≈2.5, in good agreement with the measurements. While this work is confined to ferroelectric materials, it provides a general workflow and framework for other AFM-based mechanical measurements.