Spring constants of composite ceramic/gold cantilevers for scanning probe microscopy

Abstract

A combination of finite element analysis (FEA) calculations and resonant frequency measurements are applied for determining the normal and lateral spring constants of microfabricated ceramic/gold cantilevers for friction force mode of scanning probe microscopy (SPM). The cantilever Si3N4 and Au layers are combined analytically into an equivalent single composite layer. Bending and torsion behavior of the cantilever are determined through FEA. Effective Young’s modulus for the composite Si3N4-Au beam is determined through assimilation of FEA and fundamental resonant frequency measurements. Several current analytical solutions are compared with the full FEA evaluation. A new analytical expression is derived for obtaining the ratio of lateral to normal spring constants and thereby the evaluation of absolute values of friction coefficients. Calibration plots are presented for the assessment of both vertical and torsion spring constants of bi-component cantilevers by measuring their resonant frequencies and thickness of gold overlay. The complex relationship between resonant frequency, fo, and spring constants, k, can be presented in scaling form k∼foα with α≈2.60 for a wide range of compositions of cantilever beams. Deviations from the simple cubic relationship k∼fo3 known for a homogeneous beam are related to the bi-component nature of cantilevers with ceramic layer and gold overlay. The calibration plots are proposed and analytical expression is derived to make more accurate assessment of spring constants for composite SPM cantilevers.