REVIEW OF SCIENTIFIC INSTRUMENTS 86, 013701 (2015)
Precise and direct method for the measurement of the torsion spring constant of the atomic force microscopy cantilevers a) D. M. Jarzabek ˛
Institute of Fundamental Technological Research, Pawi´nskiego 5b, 02-106 Warsaw, Poland
(Received 2 September 2014; accepted 8 December 2014; published online 30 December 2014) A direct method for the evaluation of the torsional spring constants of the atomic force microscope cantilevers is presented in this paper. The method uses a nanoindenter to apply forces at the long axis of the cantilever and in the certain distance from it. The torque vs torsion relation is then evaluated by the comparison of the results of the indentations experiments at different positions on the cantilever. Next, this relation is used for the precise determination of the torsional spring constant of the cantilever. The statistical analysis shows that the standard deviation of the calibration measurements is equal to approximately 1%. Furthermore, a simple method for calibration of the photodetector’s lateral response is proposed. The overall procedure of the lateral calibration constant determination has the accuracy approximately equal to 10%. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904866]
INTRODUCTION
Atomic force microscope (AFM) is one of the most important instruments for measurements at nano- and microscales. It is particularly useful in the evaluation of mechanical properties of materials. Force-distance curves using AFM have been performed on a variety of different systems, ranging from immensely soft and compliant to hard and stiff.1,2 Therefore, a lot of calibration procedures for precise AFM measurements have been proposed.3–6 Furthermore, because of its high sensitivity, lateral force microscopy (LFM), which is one of the modes of AFM, allows to evaluate frictional properties in the nanoscale.2,7–9 These properties are particularly important in the nanotechnology due to the fact that as the size of mechanical devices shrinks, friction, adhesion, and wear start to play an essential role in the design of such mechanisms. Basically, in a typical LFM experiment, the cantilever moves across the sample surface along a line forming a 90◦ angle with the cantilever long axis. This technique also allows to investigate bending and fracturing of nanopillars, nanorods, etc.10 Unfortunately, the lack of a precise method of calibration of the lateral force in LFM measurements reduces its possible applications and causes that this technique does not provide accurate results. Much research in recent years has focused on the development of methods for calibrating lateral forces in LFM.11–22 A detailed review of these methods was provided by Munz23 or by Pettersson et al.24 The two approaches can be distinguished. In the first approach, the torsional spring constant of the cantilever and the lateral sensitivity of the position-sensitive photodiode are calibrated separately. The most often used method to obtain the torsional spring constant is to use calculations based on dimensions and material properties25 or the finite element analysis.26 The
a)[email protected]
0034-6748/2015/86(1)/013701/6/$30.00
photodiode calibration is usually more problematic. The photodiode’s calibration constant can be evaluated from a quantitative interpretation of the slope as the surfaces stick in the friction loop.13 Second, more common, approach is to obtain the calibration constant (the direct relation between the lateral force and the photodiode signal) by analyzing the lateral response when scanned across specific samples or calibration gratings. The most popular method of this kind is the wedge method, which has been developed by Ogletree et al.11 and improved by Varenberg et al.14 In this method, the calibration constant is determined from the analysis of the lateral response of the AFM cantilever while scanning the commercially available calibration grating TGF11. Another one is so called a pivot method.27,28 In this approach, the tip glued to the surface is pointed upwards. This is called a pivot. It approaches then to the calibrated cantilever through an upward piezoscanner movement, and it twists this cantilever. The deflection, lateral signal, and piezo z signal are recorded, and the calibration constant is then determined from the data. All methods have their advantages and disadvantages. For example, methods based on theoretical calculations from dimensions and material’s properties are simple but very inaccurate, especially in the case of the cantilevers coated with reflecting or other layers or cantilevers made of an anisotropic material, i.e., silicon. The precise measurement of the dimensions is also challenging. Furthermore, the photodetector calibration methods and the methods for determining the overall calibration constant, usually, are based on a simple model which does not take into consideration all sophisticated tribological and mechanical effects. Munz, in his review,23 describes these problems in detail. Hence, it is extremely important to establish a precise method for calibration of the lateral force in LFM. This paper proposes such a method based on the first approach. First, the torsional spring constant is determined using the accurately calibrated nanoindentation device. Second, the photodiode response is calibrated from a quantitative interpretation of
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the lateral signal while scanning the samples with steps of different heights. Finally, the results are compared with the theoretical spring constant’s values.
NORMAL AND TORSIONAL STIFFNESS CALIBRATION
The calibration procedure presented in this paper was inspired by the method, which uses the accurately calibrated nanoindenter, to determine the normal stiffness of an investigated cantilever.29 However, in our case, the loads were applied not only at the long axis of the cantilever but also at a certain distance from it (Fig. 1(a)). The cantilever indented close to its edge not only deflects but also twists. Hence, it is possible to determine the torsion vs force relation (Fig. 1(b)). The experiments were carried out with a CSM Ultra Nanoindentation Tester. The device includes an XYZ sample stage, which is used to precisely position the sample under the indenter and the optical microscope. The accuracy of the positioning equals 0.25 µm. First, the cantilever is put on a flat and smooth table. Its position is controlled by the optical microscope. Its longer axis is parallel to the surface of the table and perpendicular to the direction of the applied force. The cantilever is then carefully positioned under the indenter tip. The diamond Berkovich tip was used. The measuring head consists of a measuring axis and a reference axis. Each has its own actuating means, displacement measuring system, and
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applied force measuring means. The indentation measurement can therefore be performed relative to a reference, whereof the application force on the cantilever is accurately controlled. The independent control of the axis of reference makes it possible to prevent any disturbance of the sample and/or of the measuring instrument itself (i.e., resulting from a temperature variation). The load application is done by a piezoactuator. The highest load resolution which can be achieved is 10 nN. Displacement is measured by a differential capacitive sensor with 0.001 nm resolution. Typical values of the thermal drift are less than 0.05 nm/min. The measurements were carried out at room temperature. The variation of the temperature should be less than 1 ◦C. In order to determine the normal stiffness of a cantilever, 10 force-displacement curves were carefully performed at the long axis of the cantilever. The distance from the end of the cantilever was higher than 30 µm concerning the position of the cantilever’s tip (Fig. 1(a)). Next, 10 curves were performed at the same distance from the end of the cantilever but close to one of the cantilever’s edges. The position of the applied load was accurately measured. After that, for each indentation, the regression coefficients were evaluated by the simple linear regression. Next, the average regression coefficients for the indentation at the long axis of the cantilever (normal spring constant knor) and close to the cantilever’s edge (spring constant k) were determined. The normal spring constant knor and the torsional spring constant k tor are defined as knor =
F d
k tor =
F , t
(1)
where F is the applied load and d and t are deflection and twist of the cantilever, respectively. In order to determine ktor, it was assumed that the cantilever, during indentation close to its edge, behaves like two springs in series. Hence 1 1 1 = − . (2) ktor k knor On the other hand, the torsional spring constant, α, can be described by the following equation M , (3) γ where M stands for the applied torque and γ for the angle of twist. The angle of twist is equal to α=
t tgγ = , (4) r where r stands for the distance between the measurement position and the central long axis of the cantilever. Therefore, for small values t γ= . (5) r Finally, FIG. 1. Torsional spring constant calibration procedure. (a) The indentation across the cantilever. The loads were applied not only at the long axis of the cantilever but also at a certain distance from it. The cantilever indented close to its edge not only deflects but also twists; (b) An example of the force vs distance relation.
M F · r F · r2 = = = k tor · r 2. (6) γ γ t From the indentation experiments, it is only possible to determine the torsional and normal spring constants exactly α=
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at the position of the applied load. In order to determine these constants at the position of the cantilever’s tip, the beam theory should be used. According to the beam theory, normal and torsional spring constants of the cantilever can be evaluated from the following equations: knorcan =
Ewd 3 , 4L 3can
(7)
Gwd 3 , (8) 3L can where w stands for the cantilever’s width, d for the cantilever’s thickness, E for the Young’s modulus, and G for the shear modulus of the cantilever’s material. Hence, if the crosssection of the cantilever does not change along the cantilever, then the relation between 1/α and the distance from the base of the cantilever to the position of the measurement l is linear αcan =
1 ∼ l. (9) α According to this relation, it is possible to evaluate the value of α at the position of the cantilever’s tip. The torsional spring constant of the cantilever αcan can be calculated by L mes , (10) L can where αcan stands for the measured torsional spring constant (torsional spring constant at the distance from the base of the cantilever to the measurement point), L mes stands for the distance from the base of the cantilever to the measurement point, and L can stands for the distance from the base of the cantilever to the cantilever’s tip. Moreover, the normal spring constant, knorcan, of the cantilever should, in this case, be transformed in line with the following equation: αcan = αmes
L 3mes , (11) L 3can where k normes stands for the measured normal spring constant. The example of the above relations for an investigated cantilever is shown in Fig. 2. It is evident that 1/α and knor−1/3 are proportional to l. Table I shows the results of the measurements and the theoretical spring constants of two investigated, uncoated, rectangular cantilevers made of silicon. The dimensions of the cantilevers were measured using scanning electron microscope with accuracy equal to 0.05 µm (Fig. 3). It should be noted that, usually, the cantilevers are not perfectly rectangular. For example, it is common to use cantilevers with trapezoidal cross-section (i.e., Fig. 1(a)). The results of the measurement of such a cantilever are also presented in Table I. The theoretical calculations were not done for this case. On the other hand, the calibration procedure described in this paper is not influenced by the shape of the cross-section of a cantilever. k norcan = knormes
THE ACCURACY OF THE TORSIONAL STIFFNESS MEASUREMENT
The accuracy of positioning, according to the manufacturer of the XY table, was 0.25 µm. Therefore, the
FIG. 2. Final result of the normal (a) and torsional (b) spring constant measurement. 1/α and knor−1/3 are proportional to l. Hence, it is possible to determine the values of the constants at the cantilever’s tip by measurement of the constants at any place at the cantilever.
procedure is more accurate for wider cantilevers. The accuracy of the force measurement was equal to 10 nN and the displacement resolution to 0.001 nm. Due to high precision in the measurement, the inaccuracy of α estimation (the inaccuracy of the proportional constant determination) for the investigated cantilevers (Fig. 2(b), Table I) was equal to 1% or even lower. It should be noted that this inaccuracy can be higher in the case of shorter and narrower cantilevers. However, it should not exceed 5% for the commercially available cantilevers.
PHOTODETECTOR CALIBRATION
The main purpose of this paper is to present the method for the precise measurement of the torsion spring constant of the cantilevers. Nevertheless, a simple method for the lateral photodetector’s response is also presented. The commercially available calibration gratings, TGZ1 and TGZ2 (NT-MDT), were used in order to calibrate the lateral response of the photodetector. On the surface of these samples, there are steps with known height. It was assumed that if the feedback is
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TABLE I. The results of the normal and torsional spring constants evaluation. The table shows the results of the measurements and the theoretical spring constants of two investigated, uncoated, rectangular cantilevers made of silicon. The dimensions of the cantilevers were measured using a scanning electron microscope with the accuracy equal to 0.05 µm. The measurement errors are given by 1 standard deviation.
Number
Normal spring constant (beam theory) (N/m)
Normal spring constant (measurement) (N/m)
Torsional spring constant (beam theory) (µ Nm/rad)
Torsional spring constant (measurement) (µ Nm/rad)
Rectangular cross-section of the cantilever 1
116 ± 13
135 ± 1
2
2.2 ± 0.7
2.51 ± 0.05
0.5 ± 0.1
0.695 ± 0.005
0.01 ± 0.005
0.0121 ± 0.0004
Trapezoidal cross-section of the cantilever 3
2.08 ± 0.01
...
switched off and the deflection of the cantilever (normal load) is high enough, the cantilever mainly twists during contact with the step (Fig. 4)—the variations in the deflection can be omitted. The torsion angle, ϕ, for small cantilever’s lateral displacement, x, according to Fig. 4 can be described by the following equation: ϕ = arctan
x h − h s + 12 d
≈
x h − h s + 12 d
,
(12)
where h stands for the tip’s height and h s for the step’s height. The lateral photodetector’s response constant β is defined as ϕ , (13) Ud where Ud stands for the value of the photodetector’s voltage. In order to eliminate the necessity of the tip’s height and the cantilever’s thickness measurements, two calibration gratings with different step heights, h s1 = 107.6 ± 2 nm and h s2 = 560±4 nm, were used. If the cantilever is twisted by the same angle, ϕ, on both gratings then β=
x1 x2 x1 − x2 ϕ= = = , h − h s1 + 12 d h − h s2 + 21 d h s2 − h s1
0.0313 ± 0.0005
...
two different calibration gratings with step heights equal to h s1 and h s2, respectively (Fig. 5). Finally, β can be determined from the following equation: β=
x1 − x2 . Ud (h s2 − h s1)
(15)
In order to increase the accuracy of the β determination, the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement can be calculated from the simple linear regression (Fig. 6) for both gratings. It should be noted that the final results depend on the cantilever and the position of the laser spot. In the case of the first cantilever, from Table I, the lateral photodetector’s response constant β = 11.7 ± 0.6
rad . µV
(16)
Finally, the calibration constant, a, of this cantilever a = β · αcan = 8.1 ± 0.5
nN . mV
(17)
(14)
where x 1 and x 2 stands for the values of the cantilever’s lateral displacements for the twist angle, ϕ, while scanning
FIG. 3. SEM picture of a cantilever 2. In order to compare the results of the calibration with the standard beam theory, the dimensions of the cantilevers were measured using scanning electron microscope with the accuracy equal to 0.05 µm.
FIG. 4. The scheme of the photodetector’s lateral response calibration. It was assumed that if the feedback is switched off and the deflection of the cantilever (normal load) is high enough, the cantilever mainly twists during contact with the step.
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according to (12) and (13), the value of h + 1/2d can be determined from the following equation: 1 x1 x2 h+ d = + h s1 = + h s2, 2 Ud β Ud β
(18)
and for the investigated cantilever, it was equal to 15.7 ± 0.7 µm. h + 1/2d can also be measured with much higher accuracy by the scanning electron microscope and was equal to 17.51 ± 0.01 µm. The difference in these values is equal to approximately 10%. Hence, this value gives us the real inaccuracy of the method presented here. The most important challenge in the further development of this method is also the technique of an accurate photodetector’s response calibration.
DISCUSSION FIG. 5. The scheme of the difference in the cantilever’s behaviour while scanning steps with different heights. In order to eliminate the necessity of the tip’s height and cantilever’s thickness measurements, two calibration gratings with different step heights, h s1=107.6 ± 2 nm and h s2=560 ± 4 nm, were used. If the cantilever is twisted by the same angle, ϕ, on both gratings, then the photodetector’s calibration constant can be determined from the difference between the step heights and the differences in the cantilever’s lateral displacement for the same torsion angle.
THE ACCURACY OF THE PHOTODETECTOR CALIBRATION
The inaccuracy of the photodetector calibration given in (16) is a one standard deviation determined from 10 measurements. However, the method presented here does not take into consideration the imperfectness of the contact between the cantilever’s tip and the calibration grating. It is then possible that the real inaccuracy is higher. The lateral photodetector’s response constant cannot be measured with higher accuracy by any other existing method but, fortunately,
FIG. 6. The evaluation of the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement. In order to increase the accuracy of the β determination, the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement can be calculated from the simple linear regression for both gratings.
The method presented in this paper was developed in order to properly calibrate cantilevers used in nanofracture strength evaluation.10 Therefore, the method is sufficient for rather stiff cantilevers. The most important parameter, which can influence the accuracy of the measurement, is the width of the cantilever. For narrow and short cantilevers, the difference between the deflection and torsion can be difficult to measure. It is possible to use higher forces; however, it might exceed the elastic regime of cantilever’s deformation. It should be noted that in order to perform the procedure, the nanoindentation device should be used. Fortunately, many nano and microtechnology laboratories have such devices. Furthermore, the measurement can be also done using AFM with an accurately calibrated cantilever. The same approach was presented by Grutzik et al.29 in order to evaluate the normal spring constant of cantilevers. It will introduce additional measurement error, but overall measurement error should not exceed 3%, which still gives much higher accuracy than most of the other methods. The measurement error of the torsional stiffness of the cantilevers presented in this paper is approximately equal to 1%. Therefore, this procedure may be used to calibrate cantilevers in micro-electro-mechanical systems with a higher accuracy than ever before. Unfortunately, the inaccuracies in the photodetector’s response calibration significantly reduce the precision of the determination of the AFM calibration constant. The method for photodetector’s response calibration proposed in this paper introduces a measurement error approximately equal to 10%, which is comparable with other methods (i.e., modified wedge method presented by Wang and Gee22) and which unfortunately, significantly decreases the attractiveness of the overall procedure. One of the possible sources of the inaccuracy is the fact that this technique does not take the contact and the tip stiffness into consideration. The contact stiffness between the silicon tip and the silicon gratings should not affect significantly the value of the calibration constant but the tip’s stiffness, which depends on the tip height and shape, may introduce quite a significant inaccuracy. Furthermore, if the normal load is too small, the cantilever not only twists but also deflects. Therefore, the technique is suitable for AFM cantilevers with high normal
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spring constant and for the calibration of the cantilevers in micro-electro-mechanical systems. Furthermore, the method presented in this paper has some advantages over other methods. For example, Varenberg et al.’s method14,22—one of the most popular wedge methods—usually gives significantly different results of the calibration constant measurement for different loads. A satisfactory explanation for these differences has not been given yet. It should be noted that all the wedge methods have many different assumptions and may not properly describe the complex tribological effects. These effects may, however, significantly influence the final results. On the other hand, there are a few direct methods of the spring constant’s measurement, which do not have this disadvantage. The pivot method27,28 is among them, the easiest to apply, the most accurate and interesting. There are also several similarities between the pivot method and the method presented here. Basically, in both methods, the load is applied at a certain distance from the long axis of the calibrated cantilever, and the torsional spring constant is then evaluated from the forcedistance curves. The important advantage of the pivot method is that in one procedure the torsional calibration constant can be determined. However, even according to the author of pivot method, its inaccuracy is approximately equal to 15%, which is higher than the inaccuracy of the method with the use of the nanoindenter. This higher inaccuracy is due to the fact that in order to apply the pivot method, the other calibration procedure must be used to accurately measure the normal spring constant of the cantilever. Because of the fact that the procedures of the normal spring constant determination usually have the measurement error equal to about 5%, it introduces significant inaccuracy to the pivot method. Moreover, in the pivot method, the problem with proper calibration of piezoscanner arises. It causes problems with accurate determination of the position of the contact between pivot and the investigated cantilever. The reason for this inaccuracy is the flow of a piezoscanner. On the other hand, the main advantage of the method presented here is the high accuracy of the nanoindenter. This machine is an advanced device capable of performing forces equal to a few micronewtons with precisely controlled values. Therefore, the torsional stiffness of the cantilevers may be measured with the accuracy approximately equal to 1%. Such an accuracy has not been reported before. CONCLUSION
Although the problem of the accurate calibration of the photodetector still remains unsolved, in this paper, the precise method for the torsion spring constant of the rectangular cantilever’s evaluation is presented. The measurement error was estimated to be approximately equal to 1%. It is a significant improvement in comparison to other methods, especially to theoretical calculations based on the cantilever’s dimensions and material’s properties. The direct measurement
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methods do not depend on many assumptions like in the case of wedge methods, which is their biggest advantage. The nanoindenter method stands out from the other direct methods for its accuracy and simplicity due to the application of the device, which was designed for a very accurate force and displacement measurements. Furthermore, the nanoindenter method’s accuracy is not influenced by coatings. Finally, cantilevers with all shapes of cross-section can be investigated (rectangular, trapezoidal, etc.).
ACKNOWLEDGMENTS
The present research was supported by the Polish National Science Centre, Grant No. DEC-2012/07/N/ST8/03297. 1Y.
F. Dufrene, Bacteriol 184(19), 5205–5213 (2002).
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(2009). 3A. D. Slattery, A. J. Blanch, J. S. Quinton, and C. T. Gibson, Nanotechnology
24(1), 13 (2013). 4R. S. Gates, W. A. Osborn, and J. R. Pratt, Nanotechnology 24(25), 9 (2013). 5M.
Dendzik, A. Kulik, F. Benedetti, P. E. Marszalek, and G. Dietler, Nanotechnology 24(36), 6 (2013). 6A. D. Slattery, A. J. Blanch, V. Ejov, J. S. Quinton, and C. T. Gibson, Nanotechnology 25(33), 14 (2014). 7M. P. de Boer and T. M. Mayer, MRS Bull. 26(4), 302–304 (2001). 8I. Szlufarska, M. Chandross, and R. W. Carpick, J. Phys. D: Appl. Phys. 41(12), 39 (2008). 9D. Jarzabek, Z. Rymuza, T. Wada, and N. Ohmae, Int. J. Mater. Res. 99(8), 883–887 (2008). 10D. M. Jarzabek, A. N. Kaufmann, H. Schift, Z. Rymuza, and T. A. Jung, Nanotechnology 25(21), 215701 (2014). 11D. F. Ogletree, R. W. Carpick, and M. Salmeron, Rev. Sci. Instrum. 67(9), 3298–3306 (1996). 12J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70(10), 3967–3969 (1999). 13R. G. Cain, S. Biggs, and N. W. Page, J. Colloid Interface Sci. 227(1), 55–65 (2000). 14M. Varenberg, I. Etsion, and G. Halperin, Rev. Sci. Instrum. 74(7), 3362–3367 (2003). 15R. J. Cannara, M. Eglin, and R. W. Carpick, Rev. Sci. Instrum. 77(5), 053701 (2006). 16W. Liu, K. Bonin, and M. Guthold, Rev. Sci. Instrum. 78(6), 063707 (2007). 17D. Choi, W. Hwang, and E. Yoon, J. Microsc. 228(2), 190–199 (2007). 18M. A. S. Quintanilla and D. T. Goddard, Rev. Sci. Instrum. 79(2), 023701 (2008). 19H. Xie, J. Vitard, S. Haliyo, S. Regnier, and M. Boukallel, Rev. Sci. Instrum. 79(3), 033708 (2008). 20R. Alvarez-Asencio, E. Thormann, and M. W. Rutland, Rev. Sci. Instrum. 84(9), 3 (2013). 21S. S. Barkley, Z. Deng, R. S. Gates, M. G. Reitsma, and R. J. Cannara, Rev. Sci. Instrum. 83(2), 6 (2012). 22H. Wang and M. L. Gee, Ultramicroscopy 136, 193–200 (2014). 23M. Munz, J. Phys. D:. Appl. Phys. 43(6), 34 (2010). 24T. Pettersson, N. Nordgren, and M. W. Rutland, Rev. Sci. Instrum. 78(9), 8 (2007). 25J. M. Neumeister and W. A. Ducker, Rev. Sci. Instrum. 65(8), 2527–2531 (1994). 26J. L. Hazel and V. V. Tsukruk, Trans. ASME, J. Tribol. 120(4), 814–819 (1998). 27G. Bogdanovic, A. Meurk, and M. W. Rutland, Colloids Surf., B 19(4), 397–405 (2000). 28K. H. Chung and M. G. Reitsma, Rev. Sci. Instrum. 81(2), 3 (2010). 29S. J. Grutzik, R. S. Gates, Y. B. Gerbig, D. T. Smith, R. F. Cook, and A. T. Zehnder, Rev. Sci. Instrum. 84(11), 113706 (2013).
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Precise and direct method for the measurement of the torsion spring constant of the atomic force microscopy cantilevers a) D. M. Jarzabek ˛
Institute of Fundamental Technological Research, Pawi´nskiego 5b, 02-106 Warsaw, Poland
(Received 2 September 2014; accepted 8 December 2014; published online 30 December 2014) A direct method for the evaluation of the torsional spring constants of the atomic force microscope cantilevers is presented in this paper. The method uses a nanoindenter to apply forces at the long axis of the cantilever and in the certain distance from it. The torque vs torsion relation is then evaluated by the comparison of the results of the indentations experiments at different positions on the cantilever. Next, this relation is used for the precise determination of the torsional spring constant of the cantilever. The statistical analysis shows that the standard deviation of the calibration measurements is equal to approximately 1%. Furthermore, a simple method for calibration of the photodetector’s lateral response is proposed. The overall procedure of the lateral calibration constant determination has the accuracy approximately equal to 10%. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904866]
INTRODUCTION
Atomic force microscope (AFM) is one of the most important instruments for measurements at nano- and microscales. It is particularly useful in the evaluation of mechanical properties of materials. Force-distance curves using AFM have been performed on a variety of different systems, ranging from immensely soft and compliant to hard and stiff.1,2 Therefore, a lot of calibration procedures for precise AFM measurements have been proposed.3–6 Furthermore, because of its high sensitivity, lateral force microscopy (LFM), which is one of the modes of AFM, allows to evaluate frictional properties in the nanoscale.2,7–9 These properties are particularly important in the nanotechnology due to the fact that as the size of mechanical devices shrinks, friction, adhesion, and wear start to play an essential role in the design of such mechanisms. Basically, in a typical LFM experiment, the cantilever moves across the sample surface along a line forming a 90◦ angle with the cantilever long axis. This technique also allows to investigate bending and fracturing of nanopillars, nanorods, etc.10 Unfortunately, the lack of a precise method of calibration of the lateral force in LFM measurements reduces its possible applications and causes that this technique does not provide accurate results. Much research in recent years has focused on the development of methods for calibrating lateral forces in LFM.11–22 A detailed review of these methods was provided by Munz23 or by Pettersson et al.24 The two approaches can be distinguished. In the first approach, the torsional spring constant of the cantilever and the lateral sensitivity of the position-sensitive photodiode are calibrated separately. The most often used method to obtain the torsional spring constant is to use calculations based on dimensions and material properties25 or the finite element analysis.26 The
a)[email protected]
0034-6748/2015/86(1)/013701/6/$30.00
photodiode calibration is usually more problematic. The photodiode’s calibration constant can be evaluated from a quantitative interpretation of the slope as the surfaces stick in the friction loop.13 Second, more common, approach is to obtain the calibration constant (the direct relation between the lateral force and the photodiode signal) by analyzing the lateral response when scanned across specific samples or calibration gratings. The most popular method of this kind is the wedge method, which has been developed by Ogletree et al.11 and improved by Varenberg et al.14 In this method, the calibration constant is determined from the analysis of the lateral response of the AFM cantilever while scanning the commercially available calibration grating TGF11. Another one is so called a pivot method.27,28 In this approach, the tip glued to the surface is pointed upwards. This is called a pivot. It approaches then to the calibrated cantilever through an upward piezoscanner movement, and it twists this cantilever. The deflection, lateral signal, and piezo z signal are recorded, and the calibration constant is then determined from the data. All methods have their advantages and disadvantages. For example, methods based on theoretical calculations from dimensions and material’s properties are simple but very inaccurate, especially in the case of the cantilevers coated with reflecting or other layers or cantilevers made of an anisotropic material, i.e., silicon. The precise measurement of the dimensions is also challenging. Furthermore, the photodetector calibration methods and the methods for determining the overall calibration constant, usually, are based on a simple model which does not take into consideration all sophisticated tribological and mechanical effects. Munz, in his review,23 describes these problems in detail. Hence, it is extremely important to establish a precise method for calibration of the lateral force in LFM. This paper proposes such a method based on the first approach. First, the torsional spring constant is determined using the accurately calibrated nanoindentation device. Second, the photodiode response is calibrated from a quantitative interpretation of
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the lateral signal while scanning the samples with steps of different heights. Finally, the results are compared with the theoretical spring constant’s values.
NORMAL AND TORSIONAL STIFFNESS CALIBRATION
The calibration procedure presented in this paper was inspired by the method, which uses the accurately calibrated nanoindenter, to determine the normal stiffness of an investigated cantilever.29 However, in our case, the loads were applied not only at the long axis of the cantilever but also at a certain distance from it (Fig. 1(a)). The cantilever indented close to its edge not only deflects but also twists. Hence, it is possible to determine the torsion vs force relation (Fig. 1(b)). The experiments were carried out with a CSM Ultra Nanoindentation Tester. The device includes an XYZ sample stage, which is used to precisely position the sample under the indenter and the optical microscope. The accuracy of the positioning equals 0.25 µm. First, the cantilever is put on a flat and smooth table. Its position is controlled by the optical microscope. Its longer axis is parallel to the surface of the table and perpendicular to the direction of the applied force. The cantilever is then carefully positioned under the indenter tip. The diamond Berkovich tip was used. The measuring head consists of a measuring axis and a reference axis. Each has its own actuating means, displacement measuring system, and
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applied force measuring means. The indentation measurement can therefore be performed relative to a reference, whereof the application force on the cantilever is accurately controlled. The independent control of the axis of reference makes it possible to prevent any disturbance of the sample and/or of the measuring instrument itself (i.e., resulting from a temperature variation). The load application is done by a piezoactuator. The highest load resolution which can be achieved is 10 nN. Displacement is measured by a differential capacitive sensor with 0.001 nm resolution. Typical values of the thermal drift are less than 0.05 nm/min. The measurements were carried out at room temperature. The variation of the temperature should be less than 1 ◦C. In order to determine the normal stiffness of a cantilever, 10 force-displacement curves were carefully performed at the long axis of the cantilever. The distance from the end of the cantilever was higher than 30 µm concerning the position of the cantilever’s tip (Fig. 1(a)). Next, 10 curves were performed at the same distance from the end of the cantilever but close to one of the cantilever’s edges. The position of the applied load was accurately measured. After that, for each indentation, the regression coefficients were evaluated by the simple linear regression. Next, the average regression coefficients for the indentation at the long axis of the cantilever (normal spring constant knor) and close to the cantilever’s edge (spring constant k) were determined. The normal spring constant knor and the torsional spring constant k tor are defined as knor =
F d
k tor =
F , t
(1)
where F is the applied load and d and t are deflection and twist of the cantilever, respectively. In order to determine ktor, it was assumed that the cantilever, during indentation close to its edge, behaves like two springs in series. Hence 1 1 1 = − . (2) ktor k knor On the other hand, the torsional spring constant, α, can be described by the following equation M , (3) γ where M stands for the applied torque and γ for the angle of twist. The angle of twist is equal to α=
t tgγ = , (4) r where r stands for the distance between the measurement position and the central long axis of the cantilever. Therefore, for small values t γ= . (5) r Finally, FIG. 1. Torsional spring constant calibration procedure. (a) The indentation across the cantilever. The loads were applied not only at the long axis of the cantilever but also at a certain distance from it. The cantilever indented close to its edge not only deflects but also twists; (b) An example of the force vs distance relation.
M F · r F · r2 = = = k tor · r 2. (6) γ γ t From the indentation experiments, it is only possible to determine the torsional and normal spring constants exactly α=
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at the position of the applied load. In order to determine these constants at the position of the cantilever’s tip, the beam theory should be used. According to the beam theory, normal and torsional spring constants of the cantilever can be evaluated from the following equations: knorcan =
Ewd 3 , 4L 3can
(7)
Gwd 3 , (8) 3L can where w stands for the cantilever’s width, d for the cantilever’s thickness, E for the Young’s modulus, and G for the shear modulus of the cantilever’s material. Hence, if the crosssection of the cantilever does not change along the cantilever, then the relation between 1/α and the distance from the base of the cantilever to the position of the measurement l is linear αcan =
1 ∼ l. (9) α According to this relation, it is possible to evaluate the value of α at the position of the cantilever’s tip. The torsional spring constant of the cantilever αcan can be calculated by L mes , (10) L can where αcan stands for the measured torsional spring constant (torsional spring constant at the distance from the base of the cantilever to the measurement point), L mes stands for the distance from the base of the cantilever to the measurement point, and L can stands for the distance from the base of the cantilever to the cantilever’s tip. Moreover, the normal spring constant, knorcan, of the cantilever should, in this case, be transformed in line with the following equation: αcan = αmes
L 3mes , (11) L 3can where k normes stands for the measured normal spring constant. The example of the above relations for an investigated cantilever is shown in Fig. 2. It is evident that 1/α and knor−1/3 are proportional to l. Table I shows the results of the measurements and the theoretical spring constants of two investigated, uncoated, rectangular cantilevers made of silicon. The dimensions of the cantilevers were measured using scanning electron microscope with accuracy equal to 0.05 µm (Fig. 3). It should be noted that, usually, the cantilevers are not perfectly rectangular. For example, it is common to use cantilevers with trapezoidal cross-section (i.e., Fig. 1(a)). The results of the measurement of such a cantilever are also presented in Table I. The theoretical calculations were not done for this case. On the other hand, the calibration procedure described in this paper is not influenced by the shape of the cross-section of a cantilever. k norcan = knormes
THE ACCURACY OF THE TORSIONAL STIFFNESS MEASUREMENT
The accuracy of positioning, according to the manufacturer of the XY table, was 0.25 µm. Therefore, the
FIG. 2. Final result of the normal (a) and torsional (b) spring constant measurement. 1/α and knor−1/3 are proportional to l. Hence, it is possible to determine the values of the constants at the cantilever’s tip by measurement of the constants at any place at the cantilever.
procedure is more accurate for wider cantilevers. The accuracy of the force measurement was equal to 10 nN and the displacement resolution to 0.001 nm. Due to high precision in the measurement, the inaccuracy of α estimation (the inaccuracy of the proportional constant determination) for the investigated cantilevers (Fig. 2(b), Table I) was equal to 1% or even lower. It should be noted that this inaccuracy can be higher in the case of shorter and narrower cantilevers. However, it should not exceed 5% for the commercially available cantilevers.
PHOTODETECTOR CALIBRATION
The main purpose of this paper is to present the method for the precise measurement of the torsion spring constant of the cantilevers. Nevertheless, a simple method for the lateral photodetector’s response is also presented. The commercially available calibration gratings, TGZ1 and TGZ2 (NT-MDT), were used in order to calibrate the lateral response of the photodetector. On the surface of these samples, there are steps with known height. It was assumed that if the feedback is
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TABLE I. The results of the normal and torsional spring constants evaluation. The table shows the results of the measurements and the theoretical spring constants of two investigated, uncoated, rectangular cantilevers made of silicon. The dimensions of the cantilevers were measured using a scanning electron microscope with the accuracy equal to 0.05 µm. The measurement errors are given by 1 standard deviation.
Number
Normal spring constant (beam theory) (N/m)
Normal spring constant (measurement) (N/m)
Torsional spring constant (beam theory) (µ Nm/rad)
Torsional spring constant (measurement) (µ Nm/rad)
Rectangular cross-section of the cantilever 1
116 ± 13
135 ± 1
2
2.2 ± 0.7
2.51 ± 0.05
0.5 ± 0.1
0.695 ± 0.005
0.01 ± 0.005
0.0121 ± 0.0004
Trapezoidal cross-section of the cantilever 3
2.08 ± 0.01
...
switched off and the deflection of the cantilever (normal load) is high enough, the cantilever mainly twists during contact with the step (Fig. 4)—the variations in the deflection can be omitted. The torsion angle, ϕ, for small cantilever’s lateral displacement, x, according to Fig. 4 can be described by the following equation: ϕ = arctan
x h − h s + 12 d
≈
x h − h s + 12 d
,
(12)
where h stands for the tip’s height and h s for the step’s height. The lateral photodetector’s response constant β is defined as ϕ , (13) Ud where Ud stands for the value of the photodetector’s voltage. In order to eliminate the necessity of the tip’s height and the cantilever’s thickness measurements, two calibration gratings with different step heights, h s1 = 107.6 ± 2 nm and h s2 = 560±4 nm, were used. If the cantilever is twisted by the same angle, ϕ, on both gratings then β=
x1 x2 x1 − x2 ϕ= = = , h − h s1 + 12 d h − h s2 + 21 d h s2 − h s1
0.0313 ± 0.0005
...
two different calibration gratings with step heights equal to h s1 and h s2, respectively (Fig. 5). Finally, β can be determined from the following equation: β=
x1 − x2 . Ud (h s2 − h s1)
(15)
In order to increase the accuracy of the β determination, the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement can be calculated from the simple linear regression (Fig. 6) for both gratings. It should be noted that the final results depend on the cantilever and the position of the laser spot. In the case of the first cantilever, from Table I, the lateral photodetector’s response constant β = 11.7 ± 0.6
rad . µV
(16)
Finally, the calibration constant, a, of this cantilever a = β · αcan = 8.1 ± 0.5
nN . mV
(17)
(14)
where x 1 and x 2 stands for the values of the cantilever’s lateral displacements for the twist angle, ϕ, while scanning
FIG. 3. SEM picture of a cantilever 2. In order to compare the results of the calibration with the standard beam theory, the dimensions of the cantilevers were measured using scanning electron microscope with the accuracy equal to 0.05 µm.
FIG. 4. The scheme of the photodetector’s lateral response calibration. It was assumed that if the feedback is switched off and the deflection of the cantilever (normal load) is high enough, the cantilever mainly twists during contact with the step.
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according to (12) and (13), the value of h + 1/2d can be determined from the following equation: 1 x1 x2 h+ d = + h s1 = + h s2, 2 Ud β Ud β
(18)
and for the investigated cantilever, it was equal to 15.7 ± 0.7 µm. h + 1/2d can also be measured with much higher accuracy by the scanning electron microscope and was equal to 17.51 ± 0.01 µm. The difference in these values is equal to approximately 10%. Hence, this value gives us the real inaccuracy of the method presented here. The most important challenge in the further development of this method is also the technique of an accurate photodetector’s response calibration.
DISCUSSION FIG. 5. The scheme of the difference in the cantilever’s behaviour while scanning steps with different heights. In order to eliminate the necessity of the tip’s height and cantilever’s thickness measurements, two calibration gratings with different step heights, h s1=107.6 ± 2 nm and h s2=560 ± 4 nm, were used. If the cantilever is twisted by the same angle, ϕ, on both gratings, then the photodetector’s calibration constant can be determined from the difference between the step heights and the differences in the cantilever’s lateral displacement for the same torsion angle.
THE ACCURACY OF THE PHOTODETECTOR CALIBRATION
The inaccuracy of the photodetector calibration given in (16) is a one standard deviation determined from 10 measurements. However, the method presented here does not take into consideration the imperfectness of the contact between the cantilever’s tip and the calibration grating. It is then possible that the real inaccuracy is higher. The lateral photodetector’s response constant cannot be measured with higher accuracy by any other existing method but, fortunately,
FIG. 6. The evaluation of the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement. In order to increase the accuracy of the β determination, the proportional constant between the lateral photodetector’s signal and the cantilever’s lateral displacement can be calculated from the simple linear regression for both gratings.
The method presented in this paper was developed in order to properly calibrate cantilevers used in nanofracture strength evaluation.10 Therefore, the method is sufficient for rather stiff cantilevers. The most important parameter, which can influence the accuracy of the measurement, is the width of the cantilever. For narrow and short cantilevers, the difference between the deflection and torsion can be difficult to measure. It is possible to use higher forces; however, it might exceed the elastic regime of cantilever’s deformation. It should be noted that in order to perform the procedure, the nanoindentation device should be used. Fortunately, many nano and microtechnology laboratories have such devices. Furthermore, the measurement can be also done using AFM with an accurately calibrated cantilever. The same approach was presented by Grutzik et al.29 in order to evaluate the normal spring constant of cantilevers. It will introduce additional measurement error, but overall measurement error should not exceed 3%, which still gives much higher accuracy than most of the other methods. The measurement error of the torsional stiffness of the cantilevers presented in this paper is approximately equal to 1%. Therefore, this procedure may be used to calibrate cantilevers in micro-electro-mechanical systems with a higher accuracy than ever before. Unfortunately, the inaccuracies in the photodetector’s response calibration significantly reduce the precision of the determination of the AFM calibration constant. The method for photodetector’s response calibration proposed in this paper introduces a measurement error approximately equal to 10%, which is comparable with other methods (i.e., modified wedge method presented by Wang and Gee22) and which unfortunately, significantly decreases the attractiveness of the overall procedure. One of the possible sources of the inaccuracy is the fact that this technique does not take the contact and the tip stiffness into consideration. The contact stiffness between the silicon tip and the silicon gratings should not affect significantly the value of the calibration constant but the tip’s stiffness, which depends on the tip height and shape, may introduce quite a significant inaccuracy. Furthermore, if the normal load is too small, the cantilever not only twists but also deflects. Therefore, the technique is suitable for AFM cantilevers with high normal
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spring constant and for the calibration of the cantilevers in micro-electro-mechanical systems. Furthermore, the method presented in this paper has some advantages over other methods. For example, Varenberg et al.’s method14,22—one of the most popular wedge methods—usually gives significantly different results of the calibration constant measurement for different loads. A satisfactory explanation for these differences has not been given yet. It should be noted that all the wedge methods have many different assumptions and may not properly describe the complex tribological effects. These effects may, however, significantly influence the final results. On the other hand, there are a few direct methods of the spring constant’s measurement, which do not have this disadvantage. The pivot method27,28 is among them, the easiest to apply, the most accurate and interesting. There are also several similarities between the pivot method and the method presented here. Basically, in both methods, the load is applied at a certain distance from the long axis of the calibrated cantilever, and the torsional spring constant is then evaluated from the forcedistance curves. The important advantage of the pivot method is that in one procedure the torsional calibration constant can be determined. However, even according to the author of pivot method, its inaccuracy is approximately equal to 15%, which is higher than the inaccuracy of the method with the use of the nanoindenter. This higher inaccuracy is due to the fact that in order to apply the pivot method, the other calibration procedure must be used to accurately measure the normal spring constant of the cantilever. Because of the fact that the procedures of the normal spring constant determination usually have the measurement error equal to about 5%, it introduces significant inaccuracy to the pivot method. Moreover, in the pivot method, the problem with proper calibration of piezoscanner arises. It causes problems with accurate determination of the position of the contact between pivot and the investigated cantilever. The reason for this inaccuracy is the flow of a piezoscanner. On the other hand, the main advantage of the method presented here is the high accuracy of the nanoindenter. This machine is an advanced device capable of performing forces equal to a few micronewtons with precisely controlled values. Therefore, the torsional stiffness of the cantilevers may be measured with the accuracy approximately equal to 1%. Such an accuracy has not been reported before. CONCLUSION
Although the problem of the accurate calibration of the photodetector still remains unsolved, in this paper, the precise method for the torsion spring constant of the rectangular cantilever’s evaluation is presented. The measurement error was estimated to be approximately equal to 1%. It is a significant improvement in comparison to other methods, especially to theoretical calculations based on the cantilever’s dimensions and material’s properties. The direct measurement
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methods do not depend on many assumptions like in the case of wedge methods, which is their biggest advantage. The nanoindenter method stands out from the other direct methods for its accuracy and simplicity due to the application of the device, which was designed for a very accurate force and displacement measurements. Furthermore, the nanoindenter method’s accuracy is not influenced by coatings. Finally, cantilevers with all shapes of cross-section can be investigated (rectangular, trapezoidal, etc.).
ACKNOWLEDGMENTS
The present research was supported by the Polish National Science Centre, Grant No. DEC-2012/07/N/ST8/03297. 1Y.
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