Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers
Introduction
Atomic force microscopy (AFM) has been widely used in many scientific researches. It provides researchers a way to measure the mechanical properties of nanostructures. In order to improve the accuracy and repeatability of experimental data, many operating parameters of the measuring instrument should be calibrated carefully [1]. One of these parameters is the spring constant of the AFM cantilever, which affects significantly the force measurement results. In the past 10 years, many methods have been developed to obtain the spring constant of cantilevers. Theoretical methods were used to conveniently obtain the spring constant of AFM cantilevers. Gibson et al. [2] measured the change of resonant frequency and Q-factor to obtain the spring constant and gold coating thickness of cantilevers by two equations. Sader and White [3], Sader [4] and Numeister and Ducker [5] derived several equations for calculating the spring constant of cantilevers with regular geometries based on the beam theory or plate theory. The finite element method was also used to numerically obtain the spring constant of AFM cantilevers. Clifford and Seah [6], Müller et al. [7], Yeh et al. [8] and Chen et al. [9] used the finite element method to obtain the spring constant and deformation characteristics of AFM cantilevers. The finite element results for AFM spring constant were compared with other available calibrating methods and experimental data. Fabian et al. [10] obtained the spring constant and resonant frequency of nanoscale cantilever sensors by finite element method. The finite element results showed good agreement with experiment ones. The use of reference cantilevers provided another way to obtain the spring constant of AFM cantilevers. Cumpson et al. [11], [12] used the microfabricated array of reference springs (MARS) and lateral electrical nanobalance (LEN) devices to directly obtain the bending and lateral spring constant of AFM cantilevers, respectively.
During friction or scratch measurement processes, the loading conditions of AFM cantilevers are very complex. The AFM cantilevers tilt and are subjected to combined landing and horizontal forces [9], [13], [14]. The tilt angle and landing force affect the deformation behavior of cantilevers and induce prestress in cantilevers. To our knowledge, the spring constants of AFM cantilevers obtained from the existing theoretical methods and finite element analysis are under some particular or simplified conditions. The lateral and axial spring constants of AFM cantilever with specific tilt angles and landing forces have not been discussed before. Moreover, many AFM cantilevers are made of crystal silicon, of which the Poisson's ratio is in the range between 0.048 and 0.403 [15]. The effect of such large variation of Poison's ratio should be considered. In this work, in order to obtain the cantilever's deformation behavior precisely during the measurement process, the influences of tilt angles, landing forces and Poisson's ratio variations on the lateral and axial spring constants of rectangular and V-shaped AFM cantilevers were investigated.
Section snippets
Finite element method
The finite element method can be used to obtain the deformation characteristics of structures with complex geometries and boundary conditions. The high accuracy of the finite element method has been demonstrated in many studies [6], [8], [9], [10]. In order to evaluate the effect of Poisson's ratio variation on the spring constant of AFM cantilevers, the Young's modulus of cantilever was kept constant and Poisson's ratio varied in the range between 0.02 and 0.4.
Results and discussion
The influence of the landing force on the lateral spring constant of cantilevers without tilt angles has been studied in our previous study [9]. Besides, the effects of tilt angle and Poisson's ratio on the bending spring constant of the AFM cantilever without considering the landing force have also been discussed [9]. In the real measurement process, the cantilever has a specific tilt angle and is subjected to a landing force. The two parameters significantly affect the deformation
Conclusion
In this study, the finite element method was used to analyze the lateral and axial spring constants of AFM cantilevers in the force measurement process. The effects of tilt angle, prestress, and Poisson's ratio were considered in this study. From the finite element results, the following conclusions can be drawn:
- (1)
If the prestress is neglected, the lateral spring constants of both rectangular and V-shaped cantilevers are independent of the tilt angles.
- (2)
The lateral spring constants of rectangular
Acknowledgments
This work was supported by the National Science Council, Taiwan, the Republic of China under contract NSC 95-2221-E-007-015. We are also grateful to the National Center for High-performance Computing for Computer Time and Facilities.
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