Anal. Chem. 2007, 79, 1333-1338
Accuracy of the Spring Constant of Atomic Force Microscopy Cantilevers by Finite Element Method Bo-Yi Chen,† Meng-Kao Yeh,*,† and Nyan-Hwa Tai‡
Department of Power Mechanical Engineering and Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 30013 Taiwan, Republic of China
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. Since scanning tunneling microscopy was developed by Binnig et al. in 1982,1 three-dimensional surface images of metal can be measured by tunneling currents from a probe. Then the atomic force microscopy (AFM) was invented in 1986 by Binnig and Quate;2 this powerful equipment can image nonconductor surface topology and can measure the force between micro- and picoscales. AFM is widely used to measure the surface topography, intermolecular forces, and mechanical properties of organisms.3-6 In AFM, cantilever moves precisely by a piezosystem and the cantilever deformed angle can be measured by detecting the laser signals reflected from the cantilever surface using a photodiode. The accuracy of force measurement can be improved as long as the spring constant and deformation behavior of cantilevers are obtained precisely. The cantilever’s spring constant was reported * Corresponding author. E-mail:
[email protected]. Phone: (886)-35742595. Fax: (886)-3-5726414. † Department of Power Mechanical Engineering. ‡ Department of Materials Science and Engineering. (1) Binnig, G.; Rohrer, H.; Gerber, C. H.; Weibel, E. Appl. Phys. Lett. 1982, 40, 178-180. (2) Binnig, G.; Quate, C. F. Phys. Rev. Lett. 1986, 56, 930-933. (3) Poggi, M. A.; Gadsby, E. D.; Bottomley, L. A.; King, W. P.; Oroudjev, E, Hansma, H. Anal. Chem. 2004, 76, 3429-3444. (4) Jin, Y.; Wang, K.; Tan, W.; Wu, P.; Wang, Q.; Huang, H.; Huang, S.; Tang, Z.; Guo, Q. Anal. Chem. 2004, 76, 5721-5725. (5) Skulason, H.; Frisbie, C. D. Anal. Chem. 2002, 74, 3096-3104. (6) Ku ¨ hner, F.; Costa, L. T.; Bisch, P. M.; Thalhammer, S.; Heckl, W. M.; Gaub, H. E. Biophys. J. 2004, 87, 2683-2690. 10.1021/ac061380v CCC: $37.00 Published on Web 01/12/2007
© 2007 American Chemical Society
in different manners.7-21 Yu et al.7 calculated the normal spring constant kz of rectangular cantilevers by the following equation
kz ) Et3w/4L3 ) 3EI/L3
(1)
where E, t, w, and L are the Young’s modulus, thickness, width, and length of cantilever, respectively. I is moment of inertia of the cross-sectional area of cantilever. Poggi et al.8 studied the influence of trapezoidal cross section caused by etching process and modified eq 1 to be
kz )
(
)
3 2 2 3E t (a + 4ab + b ) 3 36(a + b) L
(2)
where a and b are the width of top and bottom surfaces of the cantilever. Sader and co-workers presented some theoretical studies for spring constants of cantilevers with regular geometries (e.g., rectangular, V-shaped, triangular and trapezoidal shapes, etc.) based on the beam or plate theories.9-11 The torsional spring constant kt of rectangular cantilevers was given by Sader9 as
kt )
{
Et3w 16(1 + υ)(L - ∆L) tanh
(L -w∆Lx6(1 - υ)) x6(1 - υ)
}
-1
w L - ∆L
(3)
where υ is the Poisson’s ratio of the cantilever and ∆L is the (7) Yu, M. F.; Lourie, O.; Dyer, M. J.; Moloni, K.; Kelly, T. F.; Ruoff, R. S. Science 2000, 287, 637-640. (8) Poggi, M. A.; McFarland, A. W.; Colton, J. S.; Bottomley, L. A. Anal. Chem. 2005, 77, 1192-1195. (9) Sader, J. E. Rev. Sci. Instrum. 2003, 74, 2438-2443. (10) Sader, J. E.; White, L. J. Appl. Phys. 1993, 74, 1-9. (11) Sader, J. E.; Larson, I.; Mulvaney, P.; White L. R. Rev. Sci. Instrum. 1995, 66, 3789-3798. (12) Neumeister, J. M.; Ducker, W. A. Rev. Sci. Instrum. 1994, 65, 2527-2531. (13) Heim, L. O.; Kappl, M.; Butt, H. J. Langmuir 2004, 20, 2760-2764. (14) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403-405. (15) Sader, J. E.; Chon, J. W. M.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967-3969. (16) Gibson, C. T.; Weeks, B. L.; Lee, J. R. I.; Abell, C.; Raiment, T. Rev. Sci. Instrum. 2001, 72, 2340-2343.
Analytical Chemistry, Vol. 79, No. 4, February 15, 2007 1333
distance from image tip to the cantilever’s free end. The moment acting on the tip equals the lateral force Fy multiplied by the tip height H, M ) FyH ) ktθy, where θy is the angle of twist. Neumeister and Ducker used the plate theory to find the V-shaped cantilever’s spring constants in three principal directions.12 Heim et al. studied the influence of the tilt angle of the rectangular cantilever according to the beam theory.13 Some methods are based on the resonant frequency of the cantilever. Cleveland et al.14 measured the resonant frequency shift before and after adding a small mass at the end of a cantilever; they found a relation between the cantilever’s spring constant and the resonant frequency as
kz ) 2π3L3wf 03xF3/E
(4)
where F and f0 are the density and resonant frequency of the cantilever without adding mass. f0 can be calculated from the normal spring constant kz, the mass of cantilever m, and the added mass M14
f0 )
x
1 2π
kz 0.2427m + M
(5)
Sader et al.15 provided another method to find the spring constant by dynamic analysis
kz ) 0.1906Ffw2LQfΓi f 2
(6)
where Ff is the density of the fluid, Qf is the quality factor in fluid, Γi is the imaginary component of the hydrodynamic function, and f is the cantilever’s resonant frequency in fluid. In order to improve the signal intensity, some high reflective layers are coated on the cantilever; the influence of these layers has been studied by Gibson et al.16 and Yeh et al.17 Cumpson et al.18-20 used microelectromechanical systems devices from which reference spring constants were easily found to calibrate the cantilever’s normal, torsional even longitudinal spring constants. The spring constants obtained by these devices had errors less than 7%. Although some parameters discussed in this paper have been published elsewhere as analytical formulas,9-13 they were derived based on the beam theory and the plate theory. Therefore, the equations had to meet the limitations, such as regular cantilever shapes, small thickness, symmetric loading conditions, small deformation, isotropic and homogeneous material, etc. Similarly, eq 1 or other methods derived from eq 18,14 can only obtain the rectangular cantilever’s normal spring constant. In real situations, the geometry and boundary conditions of the cantilever are so complicated that it is difficult to obtain the exact solution of the (17) Yeh, M. K.; Chen, B. Y.; Tai, N. H.; Chiu, C. C. Key Eng. Mater. 2006, 326-328, 377-380. (18) Cumpson, P. J.; Zhdan, P.; Hedley, J. Ultramicroscopy 2004, 100, 241251. (19) Cumpson, P. J.; Hedley, J.; Clifford, C. A. J. Vac. Sci. Technol., B 2005, 23, 1992-1997. (20) Cumpson, P. J.; Hedley, J. Nanotechnology 2003, 14, 1279-1288. (21) Stark, R. W.; Drobek, T.; Heckl, W. M. Ultramicroscopy 2001, 100, 207215.
1334
Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
Figure 1. (a) Finite element model of a type I AFM cantilever. (b) SEM image of type I AFM cantilever.
cantilever’s spring constant. Fortunately, the finite element method (FEM) has advantages in solving the cantilever problem with complex geometry and boundary conditions. Yeh et al.17 and Stark et al.21 used the finite element method to find the spring constant, deformation behavior, and resonant frequency of the cantilever; the results agreed well with the experimental ones. In this paper, the finite element method with incremental displacement-controlled scheme17,22 was used to analyze the deformation behavior of AFM cantilever, and the spring constant of AFM cantilever was obtained. The effects of material properties, Poisson’s ratio, cantilever’s geometry, image tip mass, combined loadings, and tilt angle were considered in the analysis. When AFM cantilever deforms during the force measurement process, geometric nonlinearity may occur under large loadings, such as pulling carbon nanotube ropes. The pulling force may be as high as 10 mN.23 The geometric nonlinearity and large deformation behavior of AFM cantilevers were considered in this paper. Furthermore, comparisons between the numerical results with the experimental results from Poggi et al.8 were made. FINITE ELEMENT METHOD Two types of cantilevers, (I) Nanoworld AT/CONT20 and (II) NT-MDT CS12, were used in the finite element analysis. The length, width, and thickness of type I cantilever are 440, 50, and 2 µm, and 225, 24, and 1 µm, respectively, for type II. The Young’s modulus and Poisson’s ratio of cantilevers are assumed to be 145 GPa and 0.3, respectively.7 The finite element model of type I cantilever in the analysis is shown in Figure 1a, in which 1550 shell and 247 solid elements were used. The dimensions of the AFM cantilever were measured by scanning electron microscopy (SEM) as shown in Figure 1b. Since the cantilever’s thickness is much smaller when compared with its length and width, the shell elements were used to model the cantilever and the image tip was modeled by solid elements. Each shell element has eight nodes, and each node has six degrees of freedom, including three displacements (Ux, Uy, Uz) and three rotations (θx, θy, θz). The solid element has 20 nodes; each node has 3 degrees of freedom (Ux, Uy, Uz). Based on the minimum potential energy theory, the equilibrium equation becomes22
{[K]l + [K]nl}{q}i+1 ) {F}i+1 - {F}i
(7)
where [K]l and [K]nl are the linear and nonlinear element stiffness (22) Lin, M. C.; Yeh, M. K. AIAA J. 1995, 33, 1728-1733. (23) Xie, S.; Li, W.; Pan, Z.; Chang, B.; Sun, L. J. Phys. Chem. Solids 2000, 61, 1153-1158.
Table 1. Resonant Frequency and Normal Spring Constant kz of Cantilevers Calculated by FEM and Compared with the Ones by Poggi et al.8 a resonant frequency (Hz)
dimensions (µm)b,c
error (%)
kz (N/m)
no.
L
a
b
t
exp (A)b
eq 5 (B)
FEM (C)
e1
e2
eq 2 (F)
FEM (G)
diff (%) e3
kmd
1 2 3 4 5 6 7 8 9 10
249.6 247.1 210.2 205.9 206.4 206.4 211.2 208.2 210.2 206.4
23.8 19.4 28.7 19.5 21.7 20.2 20.8 28.8 21.8 19.8
41.3 35.9 40.5 34.8 36.2 33.5 37.4 42.8 38.8 36.7
3.49 3.47 2.56 3.25 2.68 2.64 3.67 2.62 3.43 3.28
67 800 67 184 69 569 92 700 78 252 75 632 98 456 72 456 96 200 92 200
66 399 67 262 68 961 90 854 74 628 73 562 97 463 71 867 92 011 91 069
67 302 68 005 70 219 91 978 75 756 74 602 98 701 73 234 93 211 92 239
-2.06 0.11 -0.87 -1.99 -4.63 -2.74 -1.00 -0.81 -4.35 -1.22
0.73 -1.22 -0.93 0.77 3.18 1.36 -0.24 -1.07 3.10 -0.04
2.82 2.42 2.01 3.38 2.02 1.79 4.83 2.29 4.17 3.58
2.85 2.44 2.04 3.41 2.04 1.81 4.88 2.32 4.22 3.61
1.23 1.04 1.61 1.18 1.15 1.22 1.08 1.61 1.28 1.21
2.89 2.48 2.03 3.47 2.06 1.82 4.96 2.31 4.27 3.68
a AFM cantilever’s Young’s modulus is assumed as 130 GPa.8 b Results of Poggi et al.8 c a and b are the width of top and bottom surfaces, respectively. d km is the normal spring constant calculated from eq 1 using the mean width wm ) (a + b)/2, e1 ) (B - A)/A × 100%, e2 ) (C - A)/A × 100%, and e3 ) (G - F)/F × 100%.
matrices. {q}i+1 is the incremental displacement at ith step. {F}i+1 and {F}i are the force vectors at (i+1)th and ith incremental steps, respectively. RESULTS AND DISCUSSION The normal spring constants and natural frequencies of cantilevers calculated by the FEM are compared with the ones by Poggi et al.,8 as shown in Table 1. The cantilever’s Young’s modulus is assumed as 130 GPa.8 In Table 1, the present resonant frequency obtained by the finite element analysis shows a smaller average error e2 when compared with the error e1 obtained from eq 5. The difference e3 of the AFM cantilever’s spring constant between the present results and the ones from eq 2 reported by Poggi et al.8 are also listed in Table 1. In order to have a better AFM spring constant, some manufacturers used eq 1 with the assumption of mean width wm to calculate the spring constant of cantilevers with trapezoidal cross section, and the results are also included in Table 1 as km. The results show that FEM can be used to calculate the resonant frequency and spring constant of cantilevers easily and precisely. Many parameters affecting the spring constant, such as Young’s modulus, Poisson’s ratio, cantilever geometry, image tip mass, combined loadings, and tilt angle, were varied in the finite element analysis to find their influences on the cantilever’s characteristics. Thus, the deformation behavior of cantilevers with complex geometry and under loading conditions can be obtained. Effect of Material Properties. The AFM cantilever can be manufactured using silicon, silicon nitride, ceramics, or silicon with doping. Different material properties, temperature, and environments may affect the characteristics of cantilevers. Figure 2 shows that the spring constant of type I AFM cantilever increases with increasing Young’s modulus under small deformation. The spring constant is proportional to Young’s modulus of the AFM cantilever. Similar results obtained from eq 1 are also included for comparison in Figure 2. The differences between the results obtained from eq 1 and the finite element analysis are probably due to negligence of the influence from the image tip and overestimation of the cantilever’s width in eq 1. Poisson’s ratio affects the spring constant of the AFM cantilever also. Under uniaxial loading, the material with a larger Poisson
Figure 2. Spring constant of a type I AFM cantilever with increasing Young’s modulus.
ratio deforms more easily in the transverse direction than the ones with a smaller Poisson ratio. Sader et al.9-11 and Neumeister and Ducker12 studied the influence of Poisson’s ratio using equations developed from the beam and plate theories. These equations have some limitations due to the basic assumptions in theories. FEM can estimate the cantilever’s nonlinear deformation behavior as discussed in the next Effect of Cantilever Thickness section. Figure 3 shows that the normal spring constant of a type I AFM cantilever increases nonlinearly with increasing Poisson’s ratio. For Poisson ratios varying from 0.1 to 0.4, the spring constant increases 1.9%. Although the difference is small, the variation of Poisson’s ratio needs to be considered, in order to have more accurate experimental data. Effect of Cantilever Thickness. Equations 1-3 show that the spring constant of the AFM cantilever depends strongly on the cantilever thickness. The cantilever with thinner thickness is commonly used to have higher sensitivity in force measurement. The fabrication tolerance of thickness is ∼0.5 µm, which results in a spring constant error of 57.8% for a type I cantilever with 2-µm thickness. Inaccurate thickness does affect the spring constant and the geometrical nonlinear behavior of cantilevers. Figure 4a shows the force-displacement curves of a type I Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
1335
Figure 3. Spring constant of type I AFM cantilever with increasing Poisson’s ratio.
cantilever with different thicknesses. The normal spring constant can be calculated from the slope of the force-displacement curves. The thinner cantilever has higher sensitivity, smaller spring constant, and larger nonlinear deformation. The nonlinear deformation behavior was considered in the finite element analysis by including the nonlinear part of stiffness matrix [K]nl in eq 7. A normal spring constant of cantilevers proportional to the cubic of thickness can also be seen from Figure 4b for normal force of ∼200 nN. Effect of Frequency Shift with Image Tip Mass. Cleveland et al.14 and Sader et al.15 used the resonant frequency of the cantilever to calculate the spring constant of the cantilever without considering the mass of the image tip. The mass of the image tip at the free end of the cantilever plays an important role in determining the spring constant. The image tip always exists at the end of cantilever and cannot be removed. To find the normal spring constant by the resonant frequency method, at least one small mass should be added at the end of the cantilever. The frequencies of cantilever with and without the added mass were estimated as fm and f0, respectively. Here, f0 is the resonant frequency of a cantilever with image tip mass. The resonant frequency of the cantilever without image tip f′0 can be obtained by extrapolation. f′0 and eq 4 can be used to calculate the spring constant of a cantilever with image tip. The experiment to measure the frequency shift of cantilevers is difficult and time-consuming. In previous studies, f0 was directly used in eq 4;14,15 an error may occur from neglecting the image tip mass. Moreover, the torsional and longitudinal spring constants cannot be calculated from this method. For type I cantilever of length 440 µm, the image tip is like a pyramid with 2(2)1/2 µm along each side of its base and 13 µm high. From Table 2, the resonant frequency of the model with image tip mass is 2.4% lower than that without an image tip. Since the normal spring constant is proportional to the cubic and the square power of resonant frequency, as show in eq 4 and eq 5 respectively, eq 4 is more sensitive to the resonant frequency change. When using eq 4 and eq 5 to calculate the spring constant, the frequency shift due to the mass of the image tip causes differences up to 7.3 and 4.1%, as shown in Table 2. Therefore, the influence of image tip mass cannot be ignored for cantilevers with thinner thickness. 1336 Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
Figure 4. (a) Force-displacement curves and (b) normal spring constant of type I cantilever with different thicknesses.
Effect of Combined Loadings. For contact mode measurement of AFM, the image tip lands on the sample first, and a normal force is applied to the image tip at the same time. In this section, the combined loading condition of a type II AFM cantilever in lateral force measurement is investigated. In Figure 5; step 1 corresponds to the process of landing the tip onto the sample. The normal force Fz acting on the cantilever is from 0 to 1 µN. After landing the tip on the sample, the tip moves horizontally to measure the lateral force. The normal force is kept at 1 µN, and lateral force is from 0 to 10 µN in step 2. The bending angles of the cantilever at the free end with respect to the x-axis during this process are plotted in Figure 5 for different widths of cantilevers. The lateral force Fx acting in the x-direction was assumed to affect only the twist angle of cantilever θy in previous literature. However, Figure 5 shows that the lateral force Fx also affects the signal of bending angle θx to a certain extent. In lateral force measurement, the normal force should be kept constant. This indicates that if only the bending angle θx is controlled in a measurement, the normal force changes accordingly during the lateral force measurement process. The AFM cantilever’s deformation due to combined loading is so complex that some simple equations, such as eqs 1-6, cannot be used to find the AFM cantilever’s characteristics. Therefore, in order to have accurate
Table 2. Resonant Frequency Shift and Normal Spring Constant Variations Caused by Image Tip Mass of a Type I Cantilever from Eq 4, Eq 5, and Finite Element Analysisa without image tip mass
with image tip mass
kz(N/m) type I cantilever thickness (µm)
resonant freq (Hz) (J)
eq 4 (K)
0.5 1.0 1.5 2.0
3511 7022 10532 14041
0.003 38 0.027 01 0.091 14 0.215 95
a
image tip mass influence (%)
kz(N/m)
eq 5 (M)
resonant freq (Hz) (N)
eq 4 (O)
eq 5 (P)
freq e4
eq 4 e5
eq 5 e6
0.003 03 0.024 22 0.081 72 0.193 66
3430 6939 10449 13958
0.003 15 0.026 06 0.089 00 0.212 14
0.002 91 0.023 72 0.080 60 0.191 68
2.4 1.2 0.8 0.6
7.3 3.6 2.4 1.8
4.1 2.1 1.4 1.0
e4 ) (J - N)/N × 100%, e5 ) (O - K)/O × 100%, and e6 ) (M - P)/P × 100%. J and N are the resonant frequencies calculated by FEM.
Figure 5. Deformation behavior of a type II cantilever with different widths in lateral force microscopy under combined loading condition.
results, FEM is recommended to solve the problem of the deformation behavior of the AFM cantilever under combined loading. Effect of Prestress. The lateral force twists the cantilever and shifts the laser signal on the photodiode in lateral force measurement. In order to improve the accuracy of lateral force measurement, the cantilever’s torsional spring constant kt should be obtained first. Some equations9,12 can be used for kt for the cantilever with only regular geometry. Therefore, for the cantilevers with complex geometry and boundary conditions, there is no analytical formula available for kt. The lateral electrical nanobalance device, reported by Cumpson and Hedley,20 is convenient for obtaining kt but its uncertainty of measurement is ∼7%. Besides, when a cantilever lands on the sample surface, the normal force causes a prestress in the AFM cantilever, and thus affects the deformation behavior of the AFM cantilever, which leads to some errors on the subsequent measurement. The prestress in the AFM cantilever, having not been discussed before, may vary the twisting behavior of cantilevers. The normal force cannot be avoided in many experiments, such as friction measurement, surface wear test, and other measurements with combined loadings. This situation was discussed using a type II cantilever. The analysis steps of the influence of prestress in the AFM cantilever were as follows: (1) A normal force Fz is applied on the image tip. (2) The deformed profile and stress distribution of
Figure 6. Torsional spring constant of a type II cantilever in lateral force microscopy with and without prestress.
Figure 7. Spring constant of a type I cantilever with different tilt angles.
cantilever were calculated. (3) The displacements and stresses on each node were recorded. The recorded stresses were assumed as the prestress in the subsequent analysis. (4) Together with the prestress recorded in step 3, the lateral force Fx was applied on the image tip and the cantilever’s torsional spring constant with prestress can be obtained. In Figure 6, the influence of the prestress on the torsional spring constant is obvious for the AFM cantilever with and without consideration of the landing force. Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
1337
Neglecting the prestress resulting from 1 µN landing force, the torsional spring constant has errors up to 30.9% for a cantilever of width 24 µm. Therefore, the prestress should be considered in the lateral force measurement to improve the accuracy of measurement. Effect of Tilt Angle. The AFM cantilever has a tilt angle of ∼10-20°, depending on the different AFM equipment. This tilt angle also affects the spring constant kz of the cantilever. Heim et al. discussed this effect and derived an equation to calibrate the error.13 However, only the rectangular cantilever has been studied and the equation developed by Heim et al. was based on the beam theory. Therefore, the following limitations still remain: slender cantilever, regular geometry, neglecting shear stress, and only pure moment acting on the cantilever. In the finite element analysis, the spring constant kz is defined as
kz )
Fz on the image tip (N/m) Uz in the z-direction
(8)
Figure 7 shows the spring constant of a type I AFM cantilever with different tilt angles. With a 20° tilt angle, the spring constant kz is 16.1% higher when compared with that of the AFM cantilever without considering the tilt angle. CONCLUSIONS In this paper, the deformation behavior of AFM cantilevers was investigated in depth using the finite element method and spring constants of cantilevers were calculated accordingly. Many parameters, such as Young’s modulus, Poisson’s ratio, cantilever geometries, image tip mass, and tilt angle were discussed. The effects of combined loadings and prestress were first discussed
1338
Analytical Chemistry, Vol. 79, No. 4, February 15, 2007
in this paper. The finite element method can handle the problem with complex geometry and loading conditions and can overcome the limitations in the beam theory, the plate theory, and the parallel beam approximation. According to the simulation results, the following conclusions can be drawn. (1) The spring constant kz is proportional to the Young modulus of the AFM cantilever. Poisson’s ratio affects the spring constant kz up to 1.9%. (2) Thickness variation affects the deformation behavior of cantilevers, especially for the nonlinear deformation behavior of thinner cantilever, and should be considered to improve the accuracy of measurement. (3) Neglecting the mass of the image tip causes an error of spring constant kz up to 7.3%. (4) When applying combined loadings on the cantilever, the lateral force not only generates the twist angle θy of AFM cantilever but also decreases the bending angle θx of AFM cantilever. This coupling effect results in the normal force variation during the lateral force measurement. (5) Prestress, caused by normal force during the landing process, affects the torsional spring constant of cantilevers significantly, especially in the lateral force measurement. For a type II cantilever, neglecting the prestress caused by 1 µN landing force brings an error up to 30.9% in the subsequent measurements. (6) The tilt angle in AFM equipment causes an experimental error up to 16.1% for the cases considered. ACKNOWLEDGMENT This work was supported by the National Science Council, Taiwan, the Republic of China under contract NCS 94-2212-E007032. Received for review July 27, 2006. Accepted December 5, 2006. AC061380V
