Sensitivity of flexural and torsional vibration modes of atomic force microscope cantilevers to surface stiffness variations

Dynamic modes of atomic-force microscopy offer new possibilities for imaging because of the availability of the various vibrational modes of the probes. However, each mode has a different sensitivity to variations in surface stiffness. This sensitivity directly controls the image contrast. Low-stiffness cantilevers have typically been unusable for imaging of stiff materials because of the lack of sensitivity of the first flexural mode. In this paper, the sensitivities of the flexural and torsional modes are derived. Closed-form expressions are obtained for cantilevers with constant cross sections. For cantilevers with other shapes, an approximate solution is developed using the method of Rayleigh-Ritz. The interaction of the cantilever with the surface is modelled by linear springs, which restricts the results to experiments involving low-amplitude excitations. Both flexural and torsional vibration modes are considered. For given nominal values of surface and AFM probe properties, the appropriate mode for highest contrast may be predicted.