Stick-slip motion in the atomic force microscope

Measurements of atomic friction in the atomic force microscope frequently show periodic variations at the lattice spacing of the surface being scanned, which have the saw-tooth wave form characteristic of "stick-slip" motion. Simple models of this behaviour have been proposed, in which the "dynamic element" of the system is provided by the elastic stiffness and inertia of the cantilever which supports the tip of the microscope, in its lateral, i.e., torsional mode of vibration. These models have been successful in predicting the observed motion, but only by assuming that the cantilever is heavily damped. However, the source of this damping in a highly elastic cantilever is not explained. To resolve the paradox, it is shown in this note that it is necessary to introduce the elastic stiffness of the contact into the model. The relationship between the contact stiffness, the cantilever stiffness and the amplitude of the periodic friction force is derived in order for stick-slip motion at lattice spacing to be achieved.