Accuracy of the Spring Constant of Atomic Force Microscopy Cantilevers by Finite Element Method

Abstract

Atomic force microscopy (AFM) probe with different functions can be used to measure the bonding force between atoms or molecules. In order to have accurate results, AFM cantilevers must be calibrated precisely before use. The AFM cantilever's spring constant is usually provided by the manufacturer, and it is calculated from simple equations or some other calibration methods. The spring constant may have some uncertainty, which may cause large errors in force measurement. In this paper, finite element analysis was used to obtain the deformation behavior of the AFM cantilever and to calculate its spring constant. The influence of prestress, ignored by other methods, is discussed in this paper. The variations of Young's modulus, Poisson's ratio, cantilever geometries, tilt angle, and the influence of image tip mass were evaluated to find their effects on the cantilever's characteristics. The results were compared with those obtained from other methods.

Nanomechanical resonators and their applications in biological/chemical detection: Nanomechanics principles

“…To capture mode coupling in the high-frequency behavior in an AFM cantilever, it is important to include terms related to torsional motion in the equation of motion. It is generally difficult to analytically solve the equation of motion for such a case, while it can be computationally tractable using finite element methods [73]. The topic of mode-coupling in the vibrational dynamics of micro/nanocantilevers has been recently received special attention because the mode-coupling has been found to improve the AFM imaging quality [74][75][76].…”

Quantitative Normal Force Measurements by Means of Atomic Force Microscopy Towards the Accurate and Easy Spring Constant Determination

“…Clifford et al [98] showed the modification of the solution proposed by Neumaister and Ducker [97] where match level of 1% was obtained. Also Chen et al developed this particular approach [99]. They presented the impact of various factors on the calculation uncertainty such as: Poisson factor, the cantilever's thickness variation, mass of the tip, the tilt angle and presence of the non-axial mass causing torsional forces.…”

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

“…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] …”

“…Detailed discussions on the constraint of the aforementioned equations have been made in our previous study. 11 To eliminate the limitation of regular geometry and dimensional uncertainty from fabrication, Cumpson and co-workers 13,14 used the microfabricated array of reference spring ͑MARS͒ and the lateral electrical nanobalance ͑LEN͒ devices to find the bending spring constant k z and the lateral spring constant k y of the AFM cantilever directly, and the errors of the MARS and the LEN devices are estimated at 5% and 7%, respectively.…”

“…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.…”