Nanoscale elasticity measurement with in situ tip shape estimation in atomic force microscopy

For a quantitative evaluation of nanoscale elasticity, atomic force microscopy, and related methods measure the contact stiffness (or force gradient) between the tip and sample surface. In these methods the key parameter is the contact radius, since the contact stiffness is changed not only by the elasticity of the sample but also by the contact radius. However, the contact radius is very uncertain and it makes the precision of measurements questionable. In this work, we propose a novel in situ method to estimate the tip shape and the contact radius at the nanoscale contact of the tip and sample. Because the measured resonance frequency sometimes does not depend so sensitively on the contact force as expected from the parabolic tip model, we introduced a more general model of an axial symmetric body and derived an equation for the contact stiffness. Then, the parameters in the model are unambiguously determined from a contact force dependence of the cantilever resonance frequency. We verified that this method is able to provide an accurate prediction of the cantilever thickness, the tip shape, and the effective elasticity of soft and rigid samples.