Effect of length, width, and mode on the mass detection sensitivity of piezoelectric unimorph cantilevers

We have investigated both experimentally and theoretically the resonance frequency change of a piezoelectric unimorph cantilever due to the mass loaded at the tip of the cantilever. The piezoelectric cantilever was composed of a lead zirconate titanate (PZT) layer and a stainless steel layer. The dependence of the resonance frequency shift, Δf, with respect to a loaded mass, Δm, on the cantilever length, L, width, w, and the resonance modes was examined. For Δm much smaller than the effective mass of the cantilever, we showed that Δf/Δm increased with an increasing eigen value, νn2, and decreasing length, L, and decreasing width, w, as Δfn/Δm=−(νn2/4π)(1/L3w)(1/0.23612ρ̃)Ẽ/ρ̃, where Ẽ and ρ̃ are the effective Young’s modulus and effective density of the unimorph cantilever that depends on the thickness fraction, Young’s modulus and density of each of the individual layers. Thus, given the same Ẽ and ρ̃ by maintaining the same layer thickness fractions of the individual layers, Δf/Δm is increased by a factor of α−4 when a cantilever is reduced in size by a factor α in proportion in all dimensions. We also showed that the same scaling relationship can be applied to a strip mass loaded at the tip as well as uniformly distributed mass on the cantilever surface provided that the uniformly distributed mass does not form a continuous solid film rigidly bonded to the cantilever surface.