Uncertainty Quanti cation in Calibration of AFM Probes Due to Non-uniform Cantilevers

For more than two decades, the Atomic Force Microscope (AFM) has provided valuable insights in nanoscale phenomena, and it is now widely employed by scientists from various disciplines. AFMs use a cantilever beam with a sharp tip to scan the surface of a sample both to image it and to perform mechanical testing. Since the AFM measures the de ection of the probe beam, one must rst nd the spring constant of the cantilever in order to estimate the force between the sample and the probe tip. Commonly applied calibration methods regard the probe as a uniform cantilever, neglecting the tip mass and any non-uniformity in the thickness along the length of the beam. This work explores these issues, recognizing that dynamic calibration boils down to nding the modal parameters of a dynamic model for a cantilever from experimental measurements and then using those parameters to estimate the static sti ness of a probe; if the modes of the cantilever do not match the expectations, for example because non-uniformity was neglected, then the calibration will be in error. This work explores the in uence of variation in the thickness of a cantilever probe on its static sti ness as well as its dynamics, seeking to determine when the uniform beam model that is traditionally employed is not valid and how one can make sure whether the model is valid from measurable quantities. In this study, the implications for two commonly applied dynamic calibration methods, the method of Sader and the Thermal Tune method, were explored. The results show that the Sader method is quite robust to non-uniformity so long as only the rst dynamic mode is used in the calibration. The Thermal Tune method gives signi cant errors for the non-uniform probe studied here.