Dimension dependence of the thermomechanical noise of microcantilevers

Abstract

Thermomechanical noise determines the lowest detection limits of microcantilever-based devices for measuring forces and surface stress variations. In this work, arrays of 334-nm-thick single-crystalline silicon microcantilevers with dissimilar lengths and widths from 50 to 500 µm and 20 to 200 µm, respectively, have been fabricated to calculate the minimal detectable force and surface stress on the basis of the measurement of the spring constant, resonance frequency, and quality factor. The calculated minimal detectable force and surface stress are of the orders of 10–15 N Hz–1/2 and 10–7 N m–1 Hz–1/2, respectively, and both follow a nonintuitive dependence on the dimensions. The minimal detectable force decreases as the cantilevers are shorter and narrower, whereas the minimal detectable surface stress decreases by making the cantilevers shorter and wider. Theoretical expressions of the minimal detectable force and surface stress are provided as a function of the material properties, cantilever dimensions, and quality factor, which allow us to interpret the results. Both force and surface stress noises follow the same dependence on the quality factor and material properties, however, exhibit striking differences in the dimension dependences. The force and surface stress noises enhance with the quality factor. If the quality factor is kept constant, the force noise enhances as the cantilever is longer and wider, whereas the surface stress noise enhances by making the cantilever shorter and wider. The observed increase of the force noise with the length is attributed to the strong decrease of the quality factor. The results imply that the design of cantilevers for surface stress measurements in general should be different than for atomic force microscopy probes.