Position-dependent mass responsivity of silicon MEMS cantilevers excited in the fundamental, two-dimensional roof tile-shaped mode

The potential of a special class of eigenmodes of microcantilevers, the so-called roof tile-shaped modes, is investigated for resonant mass sensing. Due to the 2D shape of vibration, the mass responsivity (resonance frequency shift per unit mass change) of these modes depends on the position along the length and width of the cantilever. This is in contrast to the commonly used 1D eigenmodes, where the mass responsivity is only dependent on one spatial variable. We mapped the position-dependent mass responsivity of the fundamental roof tile-shaped mode by placing silicon dioxide microbeads as test mass on defined positions across the cantilever surface and measuring the corresponding shift in resonance frequency. The results show very good agreement with finite element analyses and are used to verify an analytic expression for the position-dependent mass responsivity. This expression allows for the calculation of the mass responsivity of an arbitrary point on the cantilever surface. A mass responsivity of 24 Hz ng−1 was found at the point of maximum oscillation amplitude at the free end of the cantilever, which rapidly decreases towards the nodal lines of the vibration. The 2D spatial distribution of the mass responsivity is discussed in this paper, which has to be considered in the design of high precision resonant mass sensors when targeting the roof tile-shaped mode. Since the accuracy in mass determination relies on the precise knowledge of the added mass position, the maximum error due to limited spatial accuracy is determined and the mid-section of the free end is identified to be the most suitable position on the cantilever surface for mass detection.