Improving picogram mass sensitivity via frequency doubling in coupled silicon micro-cantilevers

Two geometrically different cantilevers, with primary frequencies of 182.506 kHz (u-shaped cantilever for sensing) and 372.503 kHz (rectangular cantilever for detecting), were coupled by two symmetrical coupling overhangs for oscillation-based mass sensing verification with phase-locking. Based on a lumped element model, a theoretical expression, containing a nonlinear spring constant and a term corresponding to the effect of the coupling spring, was proposed to consider the factors influencing the entrainment range, which is defined as a plateau with a frequency ratio (resonant frequency of rectangular cantilever to that of u-shaped cantilever) of 2.000 in present study. A picogram order mass sensing by applying a polystyrene microsphere as a small mass perturbation onto the tip of the u-shaped cantilever was demonstrated. By varying driving voltages, two entrainment regions with and without microsphere were experimentally measured and comparatively shown. At a driving voltage of 1 Vpp, when the u-shaped cantilever was excited at its shifted frequency of 180.29 kHz, the frequency response of the coupled rectangular cantilever had a peak at double the shifted frequency of 360.58 kHz of the u-shaped cantilever. The frequency shift for picogram mass sensing was thus doubled from 2560 Hz to 5133 Hz due to phase-locking. A mass of 3.732 picogram was derived based on the doubled frequency shift corresponding to a calculated mass of 3.771 picogram from measured diameter and reported density. Both experimental demonstration and theoretical discussions from the viewpoint of entrainment range elicits the possibility of increasing the mass sensitivity via phase-locking in the coupled silicon micro-cantilevers.