Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment

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Abstract
With an optical interferometer, the free vibration spectra and the local vibration amplitude of four rectangular atomic force microscope cantilevers made of silicon have been examined experimentally in a spectral range of 100 kHz to 10 MHz. A good agreement with the flexural wave theory of elastic beams was found. Coupling to torsional vibrations was also observed. When the sensor tip of the cantilever is in contact with a sample surface the resonances are shifted in frequency and the vibration amplitudes along the cantilever change. A method is presented to calculate this frequency shift using a linear approximation for the tip–sample interaction forces, and the results are compared with the frequency shift calculated from the point‐mass model. The measured resonance frequencies of a surface‐coupled cantilever do not correspond as well to the theoretical ones as in the free case even if the elastic‐beam model is used. The reason for the disagreement is found to be the geometry of the commercial cantilevers and the nonlinearity of the tip‐sample interaction force.