Nanomechanical properties of piezoresistive cantilevers: Theory and experiment

Concise analytical expressions for the effective spring constant, resonance frequency, and effective mass are derived using elementary beam theory for cantilevers, which exhibit a basic rectangular cross section modified by a rectangular hole centered at the base, which is a typical case for piezoresistive cantilevers. The results demonstrate that each of these mechanical properties can be represented as the property of the same cantilever in the absence of the hole times a dimensionless function of the hole geometry. The derivations are used to determine the mass sensitivity of the modified levers and to optimize the mass sensitivity of the piezoresistive geometry. The beam theories are shown to be consistent with two dimensional plate theory (finite element analysis) and in good agreement with our experimental results on cantilevers milled using a focused ion beam.