High-frequency response of atomic-force microscope cantilevers

Recent advances in atomic-force microscopy have moved beyond the original quasistatic implementation into a fully dynamic regime in which the atomic-force microscope cantilever is in contact with an insonified sample. The resulting dynamical system is complex and highly nonlinear. Simplification of this problem is often realized by modeling the cantilever as a one degree of freedom system. This type of first-mode approximation (FMA), or point-mass model, has been successful in advancing material property measurement techniques. The limits and validity of such an approximation have not, however, been fully addressed. In this article, the complete flexural beam equation is examined and compared directly with the FMA using both linear and nonlinear examples. These comparisons are made using analytical and finite difference numerical techniques. The two systems are shown to have differences in drive-point impedance and are influenced differently by the interaction damping. It is shown that the higher modes must be included for excitations above the first resonance if both the low and high frequency dynamics are to be modeled accurately.