Microcantilever beams are versatile force sensors used for, among others, microaccelerometry, microelectromechanical systems, and surface force measurements, the most prominent application being atomic force microscopic imaging and force spectroscopy. Bending of the cantilever is used for simple force measurements, while changes in the amplitude or frequency of the fundamental resonance are used to detect small interaction forces or brief perturbations. Spring constants needed for quantitative measurements are determined by “reversing” the force measurements, using either Hooke’s law or the oscillation of the beam. The equality of the Hookian and the oscillating spring constant is generally assumed; however, consistent differences in experimental results suggest otherwise. In this work, we introduce a theoretical formula to describe the relationship between these two spring constants for an Euler-Bernoulli beam. We show that the two spring constants are not equal, although the percentage difference stays in the range of a single digit. We derive a general formula for the determination of effective spring constants of arbitrary eigenmodes of the cantilever beam. We demonstrate that all overtones can be treated with a linear spring - effective mass approach, where the mass remains the same for higher eigenmodes.

• Received 15 September 2008

DOI:https://doi.org/10.1103/PhysRevB.78.172101