Error in dynamic spring constant calibration of atomic force microscope probes due tononuniform cantilevers

Abstract

Many common atomic force microscope (AFM) spring constant calibration methods regardthe AFM probe as a uniform cantilever, neglecting the tip mass and any nonuniformity inthe thickness of the probe along its length. This work quantifies the error in the springconstant estimated by the Sader and thermal calibration methods due to nonuniformity inthe thickness of the cantilever and the influence of the mass loading effect of the probe tip.Formulae are presented that can be used to compute the uncertainty in cantilevercalibration for an arbitrary thickness nonuniformity, or to correct the calibration methods ifthe thickness nonuniformity is known. The results show that both methods arequite sensitive to nonuniformity. When the first dynamic mode is used in thecalibration, the error in the spring constant estimated by either method is between − 4% and 9% for a cantilever whose thickness increases or decreases linearly by 30% along itslength. The errors are several times larger if the second or higher dynamic modes areused. To illustrate the proposed methods, a commercial AFM probe that hassignificant nonuniformity is considered and the error in calibrating this probe isquantified and discussed. For this particular probe, variations in the thickness ofthe probe over the last 15% of its length are found to significantly reduce theaccuracy of the calibration when the thermal method is used, since that method issensitive to changes in the shape of the eigenmode of the probe near its free end.