Static and dynamic calibration of torsional spring constants of cantilevers.

A method for calibrating the dynamic torsional spring constant of cantilevers by directly measuring the thermally driven motion of the cantilever with an interferometer is presented. Random errors in calibration were made negligible (<1%) by averaging over multiple measurements. The errors in accuracy of ±5% or ±10% for both of the cantilevers calibrated in this study were limited only by the accuracy of the laser Doppler vibrometer (LDV) used to measure thermal fluctuations. This is a significant improvement over commonly used methods that result in large and untraceable errors resulting from assumptions made about the cantilever geometry, material properties, and/or hydrodynamic physics of the surroundings. Subsequently, the static torsional spring constant is determined from its dynamic counterpart after careful LDV measurements of the torsional mode shape, backed by finite element analysis simulations. A meticulously calibrated cantilever is used in a friction force microscopy experiment that measures the friction difference and interfacial shear strength (ISS) between graphene and a silicon dioxide AFM probe. Accurate calibration can resolve discrepancies between different experimental methods, which have contributed to a large scatter in the reported friction and ISS values in the literature to date.

Calibration of higher eigenmodes of cantilevers.

A method is presented for calibrating the higher eigenmodes (resonant modes) of atomic force microscopy cantilevers that can be performed prior to any tip-sample interaction. The method leverages recent efforts in accurately calibrating the first eigenmode by providing the higher-mode stiffness as a ratio to the first mode stiffness. A one-time calibration routine must be performed for every cantilever type to determine a power-law relationship between stiffness and frequency, which is then stored for future use on similar cantilevers. Then, future calibrations only require a measurement of the ratio of resonant frequencies and the stiffness of the first mode. This method is verified through stiffness measurements using three independent approaches: interferometric measurement, AC approach-curve calibration, and finite element analysis simulation. Power-law values for calibrating higher-mode stiffnesses are reported for several cantilever models. Once the higher-mode stiffnesses are known, the amplitude of each mode can also be calibrated from the thermal spectrum by application of the equipartition theorem.