Determination of Calibration Parameters of Cantilevers of Arbitrary Shape by Finite Elements Analysis

The use of atomic force microscopy on nanomechanical measurements requires accurate calibration of the cantilever's spring constant ($k_c$) and the optical lever sensitivity ($OLS$). The thermal method, based on the cantilever's thermal fluctuations in fluid, allows estimating $k_c$ in a fast, non-invasive mode. However, differences in the cantilever geometry and mounting angle require the knowledge of three correction factors to get a good estimation of $k_c$: the contribution of the oscillation mode to the total amplitude, the shape difference between the free and the end-loaded configurations, and the tilt of the cantilever respect to the measured surface. While the correction factors for traditional rectangular and V-shaped cantilevers geometries have been reported, they must be determined for cantilevers with non-traditional geometries and large tips. Here, we develop a method based on finite element analysis to estimate the correction factors of cantilevers with arbitrary geometry and tip dimensions. The method relies on the numerical computation of the effective cantilever mass. The use of the correction factor for rectangular geometries on our model cantilever (PFQNM-LC) will lead to values underestimated by 16%. In contrast, experiments using pre-calibrated cantilevers revealed a maximum uncertainty below 5% in the estimation of the $OLS$, verifying our approach.