Improved lateral force calibration based on the angle conversion factor ...

Journal of Microscopy, Vol. 228, Pt 2 2007, pp. 190–199 Received 18 October 2006; accepted 16 May 2007

Improved lateral force calibration based on the angle conversion factor in atomic force microscopy ∗

∗

D U K H Y U N C H O I , WO O N B O N G H WA N G & E U I S U N G YO O N † ∗

Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja Dong, Nam-gu, Pohang, Kyungbuk 790-784, Korea †Tribology Laboratory of Korea Institute of Science and Technology, Seoul, Korea

Key words. Angle conversion factor, atomic force microscopy, coefficient of friction, lateral force calibration, lateral force calibration factor, mica, twist angle. Summary A novel calibration method is proposed for determining lateral forces in atomic force microscopy (AFM), by introducing an angle conversion factor, which is defined as the ratio of the twist angle of a cantilever to the corresponding lateral signal. This factor greatly simplifies the calibration procedures. Once the angle conversion factor is determined in AFM, the lateral force calibration factors of any rectangular cantilever can be obtained by simple computation without further experiments. To determine the angle conversion factor, this study focuses on the determination of the twist angle of a cantilever during lateral force calibration in AFM. Since the twist angle of a cantilever cannot be directly measured in AFM, the angles are obtained by means of the moment balance equations between a rectangular AFM cantilever and a simple commercially available step grating. To eliminate the effect of the adhesive force, the gradients of the lateral signals and the twist angles as a function of normal force are used in calculating the angle conversion factor. To verify reliability and reproducibility of the method, two step gratings with different heights and two different rectangular cantilevers were used in lateral force calibration in AFM. The results showed good agreement, to within 10%. This method was validated by comparing the coefficient of friction of mica so determined with values in the literature. Introduction Since atomic force microscopy (AFM) (Binnig et al., 1986; Mate et al., 1987) was invented, it has received much attention from Correspondence to: Woonbong Hwang. Tel: +82 54 279 2174; fax: +82 54 279 5899; e-mail: [email protected]

researchers in surface topography, frictional properties and magnetic and other properties at the nanometer scale. Most early studies with AFM were to analyze and compare surface properties. Today, many mechanical tests (Namazu et al., 2000; Sundararajan & Bhushan, 2002; Virwani et al., 2003) are performed by AFM, and the results are compared quantitatively with experimental data obtained on the macro-scale. However, because the data measured in AFM are electronic signals for forces and surface shapes, calibration of the signals is essential ¨ for quantitative analyses (Grafstr¨om et al., 1993; Luthi et al., 1995; Putman et al., 1995). AFM lateral signal calibrations have been performed by several established methods (Ruan & Bhushan, 1994; Liu et al., 1996; Ogletree et al., 1996; Schwarz et al., 1996; Buenviaje et al., 1998; Bogdanovic et al., 2000; Cain et al., 2000; Feiler et al., 2000; Cain et al., 2001; Ecke et al., 2001; Varenberg et al., 2003a; Cannara et al., 2006). These methods determine the lateral force calibration factor by which the lateral voltage signal (V) can be converted into lateral force (N). In these calibration methods, the calibration factor is determined by means of key parameters such as the coefficient of friction, the deflection sensitivities in units of voltage signal per displacement of the tip end of the cantilever, and the contact stiffness (Carpick et al., 1997) between the cantilever tip and the sample, as shown in Fig. 1(a). For example, Ogletree et al. (1996) obtain the coefficient of friction through lateral force calibration tests on a substrate with two well-defined slopes; they then determine the lateral force calibration factor from the relations between the coefficient of friction and the gradient of lateral signals as a function of normal force. Cain et al. (2000) derive the lateral deflection sensitivity from the static portion of a friction loop between a colloidal probe and a flat surface. They then determine the calibration factor using the sensitivity and the contact stiffness. For the calibration factor, Ecke  C 2007 The Authors C 2007 The Royal Microscopical Society Journal compilation 

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Fig. 1. (a) Previous calibration methods [a: coefficient of friction, b: slope of static portion in friction loop (Carpick et al., 1997), c: normal deflection sensitivity, d: lateral deflection sensitivity, e: contact stiffness] and (b) a present calibration method to determine the lateral force calibration factor. In the previous calibration methods, the total process should be repeated for the lateral force calibration factor of different cantilevers, but in the proposed calibration method, the simple computation, which will be Eq. (10), is just required.

et al. (2001) obtain deflection sensitivities by using a colloidal probe and the cantilever with a calibrated normal spring constant. Although these are well-established methods for determining the lateral force calibration factor of an individual cantilever, they have three significant problems. The first can be called the ‘repetition problem’. The lateral force calibration factor depends on the type of cantilever, including material and dimensions. The above methods should therefore calibrate afresh for different cantilevers, as shown in Fig. 1(a). This is a critical issue in quantitative study with AFM because there are so many different cantilevers, involving distinct materials as well as dimensions. This problem will be discussed in detail in Calibration Techniques section. The second problem is uncertainty in parameters such as the deflection sensitivity and the contact stiffness. Misalignment of the photodiode or adhesion between the cantilever tip and the sample causes large variations in the parameters. The number of uncertain parameters involved in computing the calibration factor makes the uncertainty large. Liu et al. (1996) used six parameters in their calibration, and Feiler et al. (2000) used four parameters. The third problem is the complex character of the specimens. Specially prepared probes and substrates are needed for lateral force calibration in AFM. Colloidal probes were used in many cases (Cain et al., 2000, 2001; Ecke et al., 2001; Varenberg et al., 2003a; Cannara et al., 2006). Feiler et al. (2000) used probes with an attached lever assembly consisting of a glass fibre and a silica sphere. Bogdanovic et al. (2000) used a modified substrate on which a cantilever was glued with the

sharp tip pointing upwards. The preparation of these probes and substrates is not simple and leads to further statistical uncertainty. The present work deals with a novel calibration method of the lateral signal in AFM based on an angle conversion factor. The distinct calibration stages are shown in Fig. 1(b). In Step I, the angle conversion factor is determined through the moment balance equation, and then in Step II the lateral force calibration factor is obtained by elementary computation. The angle conversion factor is the invariant characteristic of an AFM system, so that, once the factor is determined, it can be applied to obtain the lateral force calibration factor of any rectangular cantilever. This method, therefore, greatly alleviates the repetition problem. To determine the angle conversion factor, the method determines the twist angles of a  cantilever via the moment balance equations ( M = 0). We focus especially on the twisting moment of a cantilever due to the lateral force in AFM. The twisting moment is responsible for the twisting of the cantilever about its length axis. The lateral force calibrations are performed on commercially available step gratings and with common rectangular cantilevers, giving a simple experimental set-up. When the angle conversion factor is calculated, the gradients of the lateral signals and the twist angles for the range of the normal force are used to eliminate the effect of the adhesion. Two step gratings with different heights and two different cantilevers are used for lateral force calibration in AFM to verify reliability and reproducibility of the proposed method. The present method is validated on the coefficient of friction of mica as measured by two different cantilevers.

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Calibration techniques Lateral force calibration factor Figure 2(a) shows the position-sensitive detector (PSD), which consists of four photosensitive sectors (or quadrants) labelled Q 1 , Q 2 , Q 3 and Q 4 . Normal or lateral forces applied to the tip of a cantilever cause the cantilever to bend or to twist, respectively. These forces change the angles of reflection of the laser, displacing the laser spot on the PSD in the vertical or horizontal direction, respectively. The PSD separately measures the normal and lateral signals, V norm = (V Q 1 + V Q 2 ) − (V Q 3 + V Q 4 ) and V lat = (V Q 1 + V Q 3 ) − (V Q 2 + V Q 4 ), which vary in proportion to the bending and twisting of the cantilever. To convert these voltage signals into units of force, the calibration factors are needed. We are concerned with the lateral force calibration factor (nN V–1 ), κ, given by F f = κ · Vlat ,

(1)

where F f is the calibrated lateral force in Newtons. The lateral force calibration factor varies according to the materials and dimensions of the cantilever used in AFM. Figure 2(b) shows the tips of two cantilevers having differing

tip heights but the same normal spring constant. When the same lateral force is given on the end of the tips, as in Fig. 2(b), the twist angle of the cantilever with the longer tip height  is larger because the moment at O is larger than that at O. The positions of the laser spot reflected on the cantilever are shown in Fig. 2(a). The reference point is the position of the reflected spots on the PSD before the cantilevers are   twisted by the lateral force, and θ 1 and θ 2 are the positions after the cantilevers are twisted. Since lateral signals in AFM are measured by the displacement of a laser spot on the PSD, the lateral signals are differently measured by the two cantilevers. Therefore, the lateral force calibration factors of two cantilevers must be different based on Eq. (1). Accordingly, whenever the different cantilever in materials and dimensions is applied in AFM, the lateral force calibration factor of the cantilever should be determined afresh. Previous calibration methods determined only the lateral force calibration factor of the cantilever used in the calibration experiments and analysis. With those methods, the experiments and analysis all have to be repeated to determine the lateral force calibration factor for different cantilevers. This is the repetition problem.

Angle conversion factor If there is no tip–surface adhesion, the lateral signal (V lat ) and the twist angle (θ t ) can be expressed by the following relations because the relation between the lateral force and the normal force, N norm is linear, and the twist angle increases linearly with the lateral signal in an AFM system (Cleveland et al., 1993; Liu et al., 1996; Schwarz et al., 1996; Buenviaje et al., 1998; Bogdanovic et al., 2000; Cain et al., 2000, 2001; Feiler et al., 2000; Ecke et al., 2001). Vlat = χ · Nnorm

(2)

θt = ξ · Nnorm ,

(3)

where χ and ξ are the gradients. An angle conversion factor, η, is defined as the ratio of the twist angle of a cantilever, θ t , to the lateral signal as follows: ηnon−ad =

Fig. 2. (a) The schematic diagram of the position-sensitive detector (PSD). The PSD consists on quadrants labelled by Q 1 , Q 2 , Q 3 and Q 4 . (b) Two cantilevers of different tip length make different twist angles under the  same lateral force. On the PSD, the laser spots move to the locations θ 1    and θ 2 by a twist of the cantilever from the reference point. θ 1 and θ 2 are then the twist displacement on the PSD shifted from the reference when the tips are twisted to θ 1 and θ 2 .

θt ξ · Nnorm ξ = = . Vlat χ · Nnorm χ

(4)

This angle conversion factor is an invariant characteristic in an AFM system based on the property of PSD. However, if there is tip–surface adhesion, as in the AFM tests, the lateral signals and the twist angles obey the following relations, in which the adhesion can be treated as an offset: Vlat = χ · Nnorm + AV

(5)

θt = ξ · Nnorm + Aθ ,

(6)

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where A V and A θ are the offset values due to adhesion. In this case, the angle conversion factor can be expressed by θt ξ · Nnorm + Aθ ηad = = . (7) Vlat χ · Nnorm + AV Equation (7) indicates that the angle conversion factor is changed by the normal force. For small normal forces, relatively large adhesions cause the angle conversion factor to be small, but for large normal forces the angle conversion factor approaches an asymptotic value. However, since the angle conversion factor is constant in an AFM system, the values calculated from Eqs (7) and (4) should be the same. When there is tip–surface adhesion, the limiting angle conversion factor is equal to the result from Eq. (4), and it becomes the ratio of the gradients of the lateral signals and the twist angles for the range of the normal force as follows: ξ · Nnorm + Aθ η = lim ηad = lim N→∞ N→∞ χ · Nnorm + AV ξ d θt /d Nnorm (8) = = ηnon−ad . = χ d Vlat /d Nnorm As a result, when there is tip–surface adhesion, the gradients (ξ ≡ dθ t /dN norm and (χ ≡ dV lat /dN norm can be applied to eliminate the effect of the adhesion in calculations of the angle conversion factor. When a lateral force is given on the tip as shown in Fig. 2(b), the moment balance at O for the small twist angle is given by  M = F f · H − kt · θ1 = 0, (9) O

where k t and H are, respectively, the torsional spring constant and the distance from the end of the tip to the centre of moment of the cantilever. By using Eqs (1), (4), (8) and (9), the lateral force calibration factor of a rectangular cantilever can be rewritten as: kt θ1 k t · =η· . (10) κ= V1 H H After the angle conversion factor is determined in an AFM system, Eq. (10) can be applied to determine the lateral force calibration factor of any rectangular cantilevers. From Eq. (10), the variables required for the lateral force calibration factor of different cantilevers are only k t and H. Theory Characteristics on the vertical step grating We used a commercially available, simple vertical step grating manufactured by the MicroMasch company (MicroMasch, Wilsonville, OR, USA), as shown in Fig. 3. Figure 3(a) is a schematic diagram and Fig. 3(b) is an AFM image of the step grating. In the schematic diagram the step is vertical. It is not vertical in AFM images, however, because of the effects of the tip radius and the side shape of the tip. This is referred to as the convolution effect of the tip. Figure 3(a) also

Fig. 3. (a) Schematic diagram and (b) AFM image of a commercially available, simple step grating. In the schematic diagram the step is vertical, but not in the AFM image due to the convolution effect in AFM. The cantilever moves in a horizontal direction to the vertical step during the lateral force calibration as indicated by the dashed line.

shows the experimental arrangement of the step grating and a rectangular cantilever used in this study. The cantilever moves in a horizontal direction to the vertical step during lateral force calibration, as indicated by the dashed line. Figure 4(a) shows the lateral signals and surface topography for a lateral force calibration on the step grating in AFM. Stages I and III show the output signals (V O ) when the tip contacts the flat surface of the step grating; Stage II is when the tip is climbing the edge of the step grating. In this study, the lateral signal (V 1 ) at the top edge of the step grating is referred to as the maximum signal. The position of the maximum signal is found to be about 40–70 nm from the top edge of the step. This difference is presumably due to the radius and the sideline of the tip. Since the tip used in this study has pyramidal shape, the sideline of the tip is almost straight; however, the end part of the tip is not straight due to the etching process that creates the sharp tip. To obtain the twist angles of a rectangular cantilever, two lateral signals (V O and V 1 ) are applied on the flat surface and at the top edge of the step grating in the moment balance equation, as shown in Fig. 4(a). We suppose for simplicity that the maximum signal occurs when the end of the tip is at the top edge of the step grating. This assumption should cause errors

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Fig. 5. Orientation of the tip for a cantilever (a) on the flat surface and (b) on the top edge of the step grating. θ 0 is the twist angle caused by the lateral force between the tip and the surface, and θ 1 is the twist angle caused by the lateral force between the end of the tip and the top edge of the step grating.

balance equation. Figure 5 shows the tip of a cantilever twisted on the flat surface and at the top edge of the step grating. When the tip is on the flat surface, the total force vector at the end of the tip is  (11) F : F = F f + N S = −μNS i + NS j, where F f is the lateral force vector, N s is the normal force vector and μ is the coefficient of friction. It has been assumed that friction depends linearly on the normal load (F f = μN s ). During the lateral force calibration, the normal force, N s , is maintained constant. The displacement vector, R, is R = −H sin θ0 i − H cos θ0 j, Fig. 4. (a) Lateral signals and topography at each step. Stages I and III indicate that the tip contacts on flat surface of the step grating, and Stage II shows that the tip is climbing on the edge of the step grating. The maximum signal (V 1 ) happens in case that the tip is at the top edge of the step grating. The real maximum signal was measured at the position deviated by about 40–70 nm from the position of the top edge of the step. (b) The real position at which the maximum signal was measured. (c) The assumed position applied to the moment balance equation. The difference of displacement vectors, R 1 and R 2 , is ignorable.

of well below 1% in the twist angles because the difference of 40–70 nm is tiny compared with the AFM tip length of 3– 15 μm, so that the difference of displacement vectors, R 1 and R 2 , in Figs 4(b) and (c) can be ignored in the calculation of the moment balance equation. To minimize the uncertainty involved, it is better to use a long sharp tip.

where H is the distance between the centre of twist and the end of a tip as shown in Fig. 5(a), and θ 0 is the twist angle on the flat surface of the step grating. The moment balance equation at point O is  M = (R × F) + T = −μH NS − H θ0 NS − kt θ0 = 0, O

(13) where k t is the torsional spring constant of the cantilever, and T is the torsional reaction of the cantilever.  The moment balance equation at point O at the top edge of the step grating can be deduced similarly. From the geometry in Fig. 5(b), the force vector at the end of the tip is given by  F : F = NT + FT = − (NT sin β + μNT cos β) i + (NT cos β − μNT sin β) j,

The twist angles of a cantilever To determine the angle conversion factor, the twist angle of the cantilever during the lateral force calibration should be obtained. Since the twist angle cannot be directly measured in AFM, this study determines the angles by means of the moment

(12)

(14)

where N T and F T are the vertical and parallel force vectors for the side surface of the tip that contacts at the top edge of the step grating. Here β is the angle between the side surface of the tip and the upper flat surface of the step grating. During the lateral force calibration, the force in the j direction is maintained

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constant as the normal force, N s . The following constraint is thereby obtained F j = NT (cos β − μ sin β) = NS .

(15) 

The displacement vector between the end of the tip and O is R = −H sin(α + β)i − H sin(α + β)j,

(16)

where α is half of the tip angle. The moment balance equation  at point O is given as follows:    M = R × F +T O

   sin β + μ cos β = −H Ns sin θ1 + cos θ1 cos β − μ sin β (17) − kt θ1 = 0,

where θ 1 is the twist angle at the top edge of the step grating. Since θ 1 is very small and α + β + θ 1 = 90◦ , Eq. (17) can be rewritten as:    cos α + μ sin α (18) = k t θ1 . −H Ns θ1 + sin α − μ cos α From Eqs (13) and (18), the twist angle of a cantilever on the flat surface of the step grating is the solution of θ02 + Ns tan α

(a − 1) Ns2 = 0, θ0 + a(b + Ns ) a(b + Ns )2

(19)

where a = V 1 /V 0 and b = kt /H . V 0 and V 1 are the lateral signals on the flat surface and at the top edge of the step grating, respectively. Here, the linear relation between lateral signals and twist angles is applied based on the definition of the angle conversion factor. Therefore, to determine the twist angle of a cantilever, we first measure the lateral signals on the step grating under a constant normal force in AFM. Then, the twist angles are finally obtained from Eq. (19) with the experimental data in AFM, the torsional spring constant and dimensions of the cantilever. Torsional spring constant of a cantilever The calibration of the torsional spring constant of the cantilever is significant to determine the lateral force calibration factor. So far, several calibration methods for the torsional spring constant have been established (Cleveland et al., 1993; Neumeister & Ducker, 1994; Carpick et al., 1997; Lantz et al., 1997; Sader et al., 1999; Green et al., 2004). In this study, the torsional spring constant of a cantilever for rectangular profile is determined from classical mechanics theory (Crandall et al., 1978; Beer & Johnston, 1992) as following: G Wt 3 , (20) 3l where G, W, l and t, respectively, denote the shear modulus, width, length and thickness of the cantilever. These dimensions kt =

195

must be measured carefully for use. In optical microscopy or scanning electron microscopy (SEM), it is possible to determine exactly the length and width of the cantilever. However, it is difficult to measure the thickness of a cantilever because the thickness of the most cantilevers used in the AFM contact mode is less than 2 μm. Since the torsional spring constant is proportional to the cube of the thickness as shown in Eq. (20), a small misalignment between the normal direction to the thickness and the view direction of microscopy causes large errors in the torsional spring constant. This study determines the thickness of a cantilever by applying the resonant frequency method of Cleveland et al. (1993). The resonant frequency method experimentally determines the normal spring constant, k N , of the rectangular cantilever. In the classical mechanics theory, the normal spring constant of the rectangular cantilever is given by kN =

E Wt 3 , 4l 3

(21)

where E is Young’s modulus of the cantilever. Through the experimentally calibrated normal spring constant, the other parameters in Eq. (21), the thickness of a cantilever can be determined. The torsional spring constant of a cantilever can be then determined from Eq. (20) with the calibrated thickness of a cantilever and the other variables. Experiments All measurements were performed in a Seiko SPA 400 AFM (Seiko, Tokyo, Japan). Common rectangular silicon nitride cantilevers fabricated by Olympus were used, and a commercially available simple silicon nitride vertical step grating, to make a simple experimental set-up. There was no further pre-treatment of cantilevers or step gratings. The crosstalk problem, due to uncertainty stemming from misalignment of the photodiode detector, is addressed before the lateral force calibration tests are performed (Ruan & Bhushan, 1994; Piner et al., 2002; Varenberg et al., 2003b). To do this, we monitor whether the normal bending of a cantilever causes lateral signals in AFM, since lateral signals should occur because of the twist in a cantilever. For the proposed calibration method, the cantilever should be carefully configured to be horizontal to the step grating, as shown in Fig. 3(a). After arranging the cantilever and the step grating in AFM, this alignment can be checked through the AFM image of the step grating measured by the configured cantilever. If the alignment between the step grating and the cantilever is well adjusted, the steps of the grating show vertically in the image. For reliability and the reproducibility of the method, two step gratings with height 100 nm and 500 nm, and two cantilevers with normal spring constants of 0.1 N m–1 (P a ) and 0.76 N m–1 (P b ), quoted by the manufacturer were used in the lateral force calibration. The length and the width of the cantilevers were measured by SEM. The thickness and the torsional spring

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Table 1. Experimental conditions. Two kinds of step gratings with different height and two rectangular cantilevers with different normal spring constants were applied

Type

Normal force (nN)

Normal spring constanta (N m–1 )

Step height (nm)

Temperature (◦ C)

Humidity (% RH)

A B C D

100–180 140–300 100–180 5–30

0.76 0.76 0.76 0.10

500 500 100 500

21 25 23 23

18 19 21 44

a From the manufacturer of Olympus Inc.

constant of the cantilever were determined from the resonant frequency method, as mentioned in the Theory section. Calibration was performed for normal loading varying from about 5 to 300 nN. Although the step grating is scanned laterally under a given normal force, the piezoelectric scanner in AFM maintains the normal force constant. Since the piezoelectric scanner may deviate from the set-point during lateral force calibration experiments, maintenance of a constant normal force should be monitored. This is easy in our experimental set-up, because the height of the step grating is always measured with the lateral signals. If the scanner deviates from the set-point, the height may be measured differently. The scan length was 5 μm, and the scan rate was 1 Hz (10 μm s–1 ). The temperature and humidity were measured. Table 1 summarizes the experimental conditions. To validate the proposed calibration method, the coefficient of friction of mica was compared with the results reported in the literature. The cantilevers, P a and P b , which were calibrated by the calibration method, were used to determine the coefficients of friction of mica. At least six tests for each cantilever were performed in different positions on mica.

Table 2. The characteristics of the rectangular cantilevers P a and P b . Length (l) and width (W) of cantilevers were measured by optical microscopy, and thickness (t) was calibrated applying the resonant frequency technique of Cleveland et al. (1993). Torsional spring constants were calculated from Eq. (20) Cantilever

l (μm)

W (μm)

t (μm)

k N (N m–1 )

k t (nNm rad–1 )

Pa Pb

180.1 86.8

36.9 36.9

0.71 0.72

0.164 1.527

2.909 6.294

to be 2.909 nNm rad–1 (P a ) and 6.294 nNm rad–1 (P b ). The characteristics of the cantilevers are summarized in Table 2. Figure 6 shows the lateral signals on the flat surface of the step grating at each normal force for each set of experimental conditions. The signals increase linearly with the normal force, although the lateral signals were shifted from the origin due to the adhesive force. As shown in the errors of Fig. 6,

140

Results and discussion 120

Lateral Signal (mV)

The normal spring constants of the cantilevers used for lateral force calibration in this study were measured as 0.164 N m–1 (P a ) and 1.527 N m–1 (P b ) using the resonant frequency method. The values of Young’s modulus and the shear modulus of low-pressure chemical vapour deposition (LPCVD) silicon nitride films prepared for commercially available cantilevers are 290 and 119 Gpa, respectively (Khan et al., 2004). From Eq. (21), the thicknesses of the cantilevers were then determined as 0.71 μm (P a ) and 0.72 μm (P a ). The calibrated thickness was very different from the thickness of 0.8 μm quoted by the manufacturer. Although the thicknesses of the cantilevers could not be measured exactly from SEM, a thickness of approximately 0.7 μm was observed from SEM images. From the material properties and dimensional data for the cantilever, the torsional spring constants were determined from Eq. (20)

100 80 60 2

Type A, R =0.9896 2 Type B, R =0.9986 2 Type C, R =0.9993 2 Type D, R =0.9901

40 20 0

0

60

120

180

240

300

360

Normal Force (nN) Fig. 6. Lateral signals versus normal force on the flat surface of the step grating. Because of the adhesive forces, the magnitude of the lateral signals is different, but the tendency of the signals is the same.

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18

197

20

To rsional Angle ( 10 rad)

2

Pa, R =0.9811

15

9 6

2

Ty pe A, R =0.9919 2 Ty pe B, R =0.9959 2 Ty pe C, R =0.9892 2 Ty pe D, R =0.9989

3

0

60

120

180

240

300

Lateral Force (nN)

-6

12

0

2

Pb, R =0.9942

16

12

360

the lateral signals measured under the same experimental conditions showed small variations due to the flat clean step grating and the consistent alignment of the photodiode. Using the experimental results, the twist angle of the cantilever on the flat surface of the step grating was determined from Eq. (19), as shown in Fig. 7. This angle also increased linearly with the normal force. In Figs 6 and 7, the gradients ξ ≡ dθ /dN and χ ≡ dV/dN were obtained, and the angle conversion factors were calculated using Eq. (8) as shown in Table 3. In the four experimental arrangements, the adhesive forces differed as a result of the different temperatures, humidities and contact positions; however, the effect of the adhesive force was eliminated by using the gradients of the lateral signals and the twist angle as a function of the normal force. As a result, the angle conversion factors showed good agreement to within 10%, indicating a mean value of 201 μrad/V. Based on Eq. (10), the corresponding lateral force calibration factors of the cantilevers P a and P b were obtained as 180 and 388 nN V–1 , respectively. To determine the angle conversion factor in Step I of the proposed calibration method, a long sharp tip should be used to prevent large errors. Since the general AFM tip has Table 3. Gradients of lateral signals and twist angles for normal force from Figs 6 and 7, and the corresponding angle conversion factors. The mean value of the angle conversion factors was 201 μrad V–1

Type

dV/dN (mV nN–1 )

dθ /dN (μrad nN–1 )

Angle conversion factor (μrad V–1 )

A B C D

0.2823 0.3533 0.2950 1.3941

0.0522 0.0700 0.0637 0.2842

185 198 216 204

4

0

Normal Force (nN) Fig. 7. Twist angles calculated at each value of the normal force on the flat surface of the step grating.

8

0

10

20

30

40

50

60

70

Normal Force (nN) Fig. 8. The calibrated lateral forces measured on mica by cantilevers P a and P b . The coefficients of friction were 0.113 for P a and 0.101 for P b .

much less tip radius than 100 nm and a tip length exceeding 3 μm, we can use the most general AFM tip in Step I. The angle conversion factor can then be applied to determine the lateral force calibration factors for any rectangular cantilever using Eq. (10) without further experiment, based on the properties of the cantilever in Step II. Equation (10) can even be applied to determine the lateral force calibration factor of the rectangular cantilevers with a colloidal tip, since all that is needed are the physical specifications of the cantilever, such as the torsional spring constant and the thickness, and the diameter of the colloidal tip for the tip length. However, the angle conversion factor cannot be applied to determine the lateral force calibration factors of triangular cantilevers, because Eq. (9) does not hold for triangular cantilevers. By using two cantilevers calibrated by the present method, the lateral forces on mica were obtained in Fig. 8. From the slopes in Fig. 8, the coefficients of friction between Si 3 N 4 tip and mica were determined to be 0.113 for cantilever P a and 0.101 for cantilever P b , showing good agreement. The coefficients of friction obtained for Si 3 N 4 and mica can be compared with the results reported in the literature. Thundat et al. (1993) reported that the coefficient of friction between mica and Si 3 N 4 cantilever having a normal spring constant of 0.06 N m–1 was 0.1 in AFM. Warmack et al. (1994) obtained a frictional coefficient of 0.086 between mica and a triangular Si 3 N 4 cantilever using a commercial AFM (Digital Instruments, Inc., Woodbury, NY, USA). Schumacher et al. (1996) measured the effect of humidity on the frictional coefficient. The coefficient of friction between mica and a triangular Si 3 N 4 cantilever was in the range 0.05–0.3, depending on the humidity. In the humidity range similar to this study, the coefficient was 0.07– 0.13. Liu et al. (1998) found that the coefficient of friction between mica and Si 3 N 4 at 20◦ C and RH 50% in AFM was

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in the range 0.06–0.1 for experimental conditions similar to the present ones. Comparison with previous results shows that the coefficients of friction of mica determined by the proposed calibration method are acceptable. This validates the proposed calibration method. Conclusions An advanced calibration method has been developed to determine lateral forces in AFM. The method separates the calibration process into two steps: the first is to determine the angle conversion factor of an AFM system, and the second is to obtain the lateral force calibration factor. By separating these steps the repetition problem is overcome. Moreover, this method applies the gradients of the lateral signals and twist angles in the calculation of the angle conversion factor. This minimizes the uncertainty due to adhesion between the tip and the sample. To permit a simple experimental set-up, this study used commercially available simple step gratings and general rectangular cantilevers. The method was validated by comparing the coefficients of friction determined for mica with values in previous reports.

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