Exact solution of the frequency shift in dynamic force microscopy

The exact frequency shift of an AFM non-uniform probe with an elastically restrained root, subjected to van der Waals force, is derived. The original distributed system is considered and then its exact fundamental solutions and the general frequency equation are derived. Results are compared with those by the force gradient method and the perturbation method. The effects of several parameters on the sensitivity of measurement are investigated. Results show that the interpretation of frequency shift by using the force gradient method is unsatisfactory. The smaller the amplitude of oscillation and the tip–surface distance are, the larger the frequency shift. The design of a taper beam is recommended for increasing the sensitivity of measurement.