Frequency equation and semi-empirical mechanical coupling strength of microcantilevers in an array

Abstract

A characteristic equation for the frequencies of the T-shaped and overhang-shaped cantilevers is derived for the first time. We show that there are optimum values of the overhang lengths and widths that maximize the frequency and the number of maxima is corresponding to the mode number. The frequency of higher-order modes could be tuned by changing the overhang dimensions. Especially, a semi-empirical formula for the coupling strength $\left(\kappa \right)$ between cantilevers in an array is proposed where the strength presents a cubic decrease with the overhang width $\left(\xi \right)$ and a linear increase with the overhang length $\left(\eta \right)$ , $\kappa =\eta /{\xi }^3$ . There is a very good agreement between the proposed formula and the values obtained in recent experiments by other researchers.

Keywords: AFM; coupling strength; frequency equation; microcantilever; overhang-shaped.