Analysis of the high-frequency response of atomic force microscope cantilevers

In most commercial atomic force microscopes, dynamic modes are now available as standard operation modes. Acoustical vibrations of atomic force microscope cantilevers can be excited either by insonification of the sample or by vibration of the clamped cantilever end. 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 such that the higher-order flexural modes are neglected. This point-mass model has been successful in advancing material property measurement techniques. The limits and validity of such an approximation have not been fully addressed. In this paper, the flexural beam equation is examined and compared with the point-mass model by 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 surface damping and the contact stiffness. The angular deflection at the end of the beam and the influence of lateral sensor tip motion are considered. It is shown that the higher modes must be included for excitations above the first resonance if both the low- and the high-frequency dynamics are to be modeled accurately.