Thermal effects on coated resonant microcantilevers

Abstract High sensitivity detectors can be made using microcantilevers and measuring shifts in the resonance frequency of the cantilever motion arising from changes in mass loading or surface stress. However, undesirable shifts in the resonance frequency can also be caused by changes of temperature, particularly for microcantilevers coated with thin films. Two problems concerning such thermal effects are addressed. One problem is the thermal mismatch-induced frequency shift due to the different thermal expansion coefficients of a coating and substrate materials. The other is the thermal drift introduced by the thermal expansion and temperature-dependent material properties. For the first problem, the equation of motion for a circular arch can be used to calculate the resonance frequency shift due to the bending caused by thermal mismatch. For the second problem, two cases are investigated, namely uniform temperature distribution and linear temperature distribution along the length of the microcantilever. It is found that the thermal frequency drift in the resonance frequency is dominated by the temperature-dependent material properties. For a typical microcantilever made from silicon, the drift is of order ∼30 ppm/°C. The thermal drift does not depend on the lever section for both uniform and linear temperature distribution cases. For coatings of different thickness, the relationship between thermal frequency drift and the ratio of coating-to-substrate thickness is nearly linear. For a cantilever coated only partly along the length, the frequency drift increases approximately linearly as a function of length of cantilever coated. The additional thermal frequency drifts arising from a thin gold coating on a typical silicon microcantilever are of order 10–30 ppm/°C. The frequency shift due to thermal mismatch bending is negligible if the temperature change is small (∼10 °C).