Error quantification 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. The AFM measures the deflection of the probe beam so one must first find 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 nonuniformity in the thickness along the length of the beam. This work explores these issues, recognizing that dynamic calibration boils down to finding the modal parameters of a dynamic model for a cantilever from experimental measurements and then using those parameters to estimate the static stiffness of a probe; if the modes of the cantilever are not what was expected, for example because the non-uniformity was neglected, then the calibration will be in error. This work explores the influence of variation in the thickness of a cantilever probe along its length on its static stiffness as well as its dynamics, seeking to determine when the uniform beam model that is traditionally employed is not valid and how one can ascertain whether the model is valid from measurable quantities. The results show that the Sader method is quite robust to non-uniformity so long as only the first dynamic mode is used in the calibration. The thermal method gives significant errors for the non-uniform probe studied here. Nomenclature αn n th frequency parameter M Moment b cantilever width ψn nth mode shape E Young’s modulus ωn natural frequency F force Q quality factor Γ hydrodynamic function ρ cantilever density h cantilever thickness ρf density of the surrounding fluid I second moment of area T temperature ks spring constant w deflection kB Boltzmann’s constant W static deflection L cantilever length χ optical lever factor m mass x coordinate along beam axis