Finite element analysis of V‐shaped cantilevers for atomic force microscopy under normal and lateral force loads

Abstract

For the quantitative investigation of surface properties such as elasticity, friction and wear by atomic force microscopy (AFM), a quantitative determination of the forces acting on the probe during its motion along the sample surface is essential. In this article, a fully parameterized finite element model for V-shaped cantilevers is presented and the three-dimensional mechanical deformations of the AFM cantilever are investigated. Force constants and detection angles for tip displacements in the three spatial directions are calculated for widely used cantilevers. The limits of linearity according to Hook's law are studied. It is found that the AFM contact cantilevers investigated here show a linear bending behavior for tip displacements within a range below approx. 10 nm in lateral directions and 100 nm in the normal direction. Displacements within this range are typical for many AFM applications including force modulation techniques. For higher loads as used e.g. for surface modification, a significant deviation from linear behavior is observed.

Effects of anisotropic material property on the spring constant and the resonant frequency of atomic force microscope cantilever

“…The accuracy of the finite element method has been verified in our previous works 11,12,16 and many other studies. 6,7,15,19 In order to obtain the deformation characteristics of the anisotropic silicon AFM cantilever accurately, the finite element method was used to analyze the spring constants and the resonant frequencies of rectangular cantilevers. The solid elements were used to model the cantilever structure, as shown in Fig.…”

“…In the published articles, researchers also used the finite element method to calculate the spring constants and deformation characteristics of AFM cantilevers. 6,7,11,12,15,16,19 In this work, two equations were proposed to describe the influence of anisotropic material property of AFM cantilevers.…”

“…Many methods were proposed to obtain the bending spring constant k z and the torsional spring constant k t of rectangular or V-shaped cantilevers. [6][7][8][9][10][11][12][13][14][15][16][17][18][19] All the existing methods used to obtain the cantilever spring constant with the shape of rectangular or V-shaped geometries assumed an equivalent isotropic property of the AFM cantilever. Based on this assumption, the bending spring constant k z of the rectangular cantilever can be expressed as 4,[6][7][8] …”

“…In general, AFM cantilevers are made by crystal silicon, 4,19 which is an anisotropic material due to its crystal lattice structure. [20][21][22][23] Manufacturers fabricate cantilevers along ͗110͘ ͑Refs.…”

Microcantilever based distance control between a probe and a surface

“…Second, the normal spring constant k Z and lateral spring constant k Y originate from the microcantilever shape, geometry and material properties. The microcantilever lateral spring constant k Y is never specified by the manufacturer but can be calculated 23,24,25,26 and is about 2·10 2 times larger than k Z for a beam shaped microcantilever vs. 1.5·10 3 times larger for a triangle shaped microcantilever (see figure 4 for microcantilever shapes).…”