Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods

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1 INSTITUTE OF PHYSICS PUBLISHING Nanotechnology 3 (2002) NANOTECHNOLOGY PII: S (02) Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods Institut Charles Sadron-Université Louis Pasteur, 6 rue Boussingault Strasbourg cedex, France levy@ics.u-strasbg.fr and maaloum@ics.u-strasbg.fr Received 3 August 200, in final form 5 November 200 Published 2 December 200 Online at stacks.iop.org/nano/3/33 Abstract Knowledge of the interaction forces between surfaces gained using an atomic force microscope (AFM) is crucial in a variety of industrial and scientific applications and necessitates a precise knowledge of the cantilever spring constant. Many methods have been devised to experimentally determine the spring constants of AFM cantilevers. The thermal fluctuation method is elegant but requires a theoretical model of the bending modes. For a rectangular cantilever, this model is available (Butt and Jaschke). Detailed thermal fluctuation measurements of a series of AFM cantilever beams have been performed in order to test the validity and accuracy of the recent theoretical models. The spring constant of rectangular cantilevers can also be determined easily with the method of Sader and White. We found very good agreement between the two methods. In the case of the V-shaped cantilever, we have shown that the thermal fluctuation method is a valid and accurate approach to the evaluation of the spring constant. A comparison between this method and those of Sader Neumeister and of Ducker has been established. In some cases, we found disagreement between these two methods; the effect of non-conservation of material properties over all cantilevers from a single chip is qualitatively invoked. (Some figures in this article are in colour only in the electronic version). Introduction The atomic force microscope (AFM) is not only an excellent tool for recording the surface topography of a sample, but thanks to recent technical advances, it has proven to be a good means studying the strength of chemical bonds [, 2], the elastic and mechanical properties of macromolecules [3 6] and the intermolecular interactions at the single molecular level [7 9]. The AFM can record the amount of force felt by the cantilever as the probe tip is brought close to and even indented into a sample surface and then pulled away. It is a great advantage, for example, to be able to derive the rupture forces involved in the experiments from an exact calibration and precise knowledge of the cantilever spring constant of each cantilever. The force sensed by the AFM probe is calculated by multiplying the deflection of the cantilever by the cantilever s spring constant. Many authors have noted that the spring values indicated by the manufacturer are incorrect [0 4]. Manufacturers of cantilevers use wide tolerances in the specified values of the force constant of cantilevers, due to the difficulties in controlling the thickness of the cantilever at a particular stage of the manufacturing process and the structural defects and deviations in geometry from cantilever to cantilever which are not taken into account. Because of this, several methods have been developed to measure or calculate the cantilever spring constant [0 26]. The most simple and elegant method used by different authors involves a study of the thermal fluctuation of the cantilever [7, 0, 6, 25, 26]. Many authors have modelled the cantilever [7, 0] as a simple harmonic oscillator with one degree of freedom (neglecting higher modes of oscillation). The potential energy of this system takes the form of 2 mω2 0 z2 = 2 K z2. The thermal method takes advantage of the equipartition theorem, applied to the cantilever potential energy: /02/ $ IOP Publishing Ltd Printed in the UK 33

2 Power spectrum density (µv 2 /Hz) Frequency (khz) Figure. Power spectrum density plot of fluctuations in the output of the cantilever deflection detector. The curve has a Lorentzian form (solid curve) around the angular resonance frequency ω f. The values of ω f, Q factor and the area below the peak are given. Figure 3. The noise power spectral density of the detector output plotted against frequency. Because of bandwidth limitation of our spectrum analyser (0 00 khz), we can see just two modes at angular resonance frequencies ω = 4. khz and ω 2 = 90. khz. Figure 2. Typical force curve, showing the deflection of a cantilever as a function of sample displacement. To avoid the hysteresis, we apply a very small voltage to the piezoelectric element in the region where the tip and the sample were in contact (see the figure inset). In this way the two slopes are completely superposed. We use this value of the slope in order to calibrate the power spectrum density. K z 2 =kt. Here, z 2 is the mean square deflection of the cantilever caused by the thermal motion in the vertical direction, k is the Boltzmann constant, T the temperature of the surrounding heat bath, K the cantilever spring constant and m the effective mass. K is deduced by measuring z 2. By transforming the time series data to the frequency domain, the Lorentzian form is obtained around the resonance frequency as shown in figure. The area below this peak is taken to be equal to the mean square z 2 of the fluctuations in the time series data; the area is indicated by z V and measured in volts. The displacement z is deduced by the relation z = C z V. In the static case (cantilever not moving), the constant C is given by the inverse of the slope, s, of the compliance region (figure 2): C = s. This argument poses three major problems: (i) We cannot take just the first peak (see figure 3), as stated by different authors, and neglect the higher modes, as this may lead to errors of about 30% in the estimation of the spring constant of the cantilever. (ii) The optical lever deflection measures the inclination dz/dx rather than the displacement (x is the coordinate in the longitudinal direction of the cantilever). Then the relation between z (nm) and z V (V) is not simply z i = z Vi /s. In order to evaluate the real displacement in each free mode, the shape of the deformation in each free mode and in the compliance region must be known. (iii) Only the first modes are accessible in the experiment (bandwidth limitation). For this reason, the repartition of the displacement is needed in the different modes to deduce z 2 and thus, the spring constant. Recently Butt and Jaschke [7] have calculated the thermal fluctuation of the rectangular cantilever using the equipartition theorem and considering all possible vibrations. They attempted to correlate dz/dx with the displacement using the optical lever technique. In this work, we have described the cantilever dynamics and thermal fluctuation behaviour of a cantilever. Thermal fluctuation measurements of a series of AFM cantilever beams have been performed. We have determined the ratio of the amplitudes of the first two modes as a function of the distance from the anchorage point, in order to test the validity and accuracy of the recent theoretical models [7]. Then we compared the experimental spring constant of the rectangular cantilevers derived from the thermal fluctuations and Sader methods [4, 5]. Next, calibration of the various shapes of cantilevers including the triangular (V-shaped) cantilevers has been determined using different methods such as thermal fluctuation, Sader Neumester and Ducker models [6] and the slope method. A comparison between these methods has been established in order to investigate the validity and accuracy of the different approaches to the evaluation of V-shaped cantilever spring constants. 2. Experimental methods All measurements were performed using a home-made AFM. The cantilevers used (figure 4) were procured from 34

3 Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods displacement Figure 5. The shape of the first and second vibration modes of a free cantilever. With our optical lever system we measured the inclination (angle) dz/dx rather than the displacement z. The x-axis represents the longitudinal direction of the cantilever. Figure 4. Cantilevers procured from Thermomicroscopes (Sunnyvale CA, USA). Thermomicroscopes (Sunnyvale CA, USA). The dimensions of the various cantilevers were determined by optical microscopy. The free vibration spectra and the vibration amplitude of the cantilevers have been examined, using a spectrum analyser (Stanford Research System), in the spectral range 0 Hz 00 KHz. The power spectrum density was recorded, in units of V 2 Hz, by taking the signal from the photodiode and feeding it directly into a spectrum analyser. For the data presented here, we calibrated each cantilever individually. With each cantilever we recorded force curves on a mica surface, using a previously calibrated piezoelectric element (performed interferometrically), immediately after collecting the thermal motion data. The slope was collected from the contact portions of the extending and retracting regions. The measurement of the slope is critical in the accurate calibration of the cantilever spring constant. As the spring constant varies with the square of the slope, an error of 0% on the slope results in an error of 20% on the spring constant. Often the force curves used from the contact portions of the extending and retraction regions present an hysteresis. To compensate for this, some authors [] have used the average value of the slopes together. Consequently, this manner to estimate the slope may lead to an error of 0% on the slope. To avoid this hysteresis, we applied a very small voltage on the piezoelectric element (very small displacement) in the region where the tip and sample were in contact (figure 2). In this way the two slopes are completely superposed. The value of the slope (V nm ) allows us to convert the power spectrum density from V 2 Hz to nm 2 Hz. Since the oscillating shape is different from the static shape, the relationship between the end position and the end angle is also different, as stated by Butt and Jaschke [7]. With our optical lever system we measure the inclination (angle) dz/dx rather than the displacement z. x is the coordinate in the longitudinal direction of the cantilever as shown in figure 5. Consequently, what we actually measure with the optical lever technique is a virtual height z, caused by the inclination dz/dx. 3. Results and discussion Rectangular cantilever. In this section, we assess the accuracy and validity of the theoretical model by presenting a comprehensive and detailed comparison with the experimental measurements. In the case of a rectangular cantilever with a free end, Butt and Jaschke [7] have shown that the virtual deflection z 2 is correlated with the real one using the following relation: z 2 4kT 4 = 3K = 3 z2. In this equation the contributions of all vibration modes have been considered. The virtual deflection is caused by the inclination dz/dx. The virtual and real deflections for each vibration mode are given by the following relations: zi 2 = 6kT 3Kαi 2 ( sin αi sinh α i sin α i + sinh α i ) 2 () zi 2 =2kT Kαi 4. (2) Here, L/α i represents the wavelength of the ith vibration mode and L is the length of the rectangular cantilever (figure 5). Consequently, the amplitude of thermal noise in the deflection falls by a factor of 00 and the virtual deflection decreases by a factor of about six when going from the first to the sixth vibration modes. The α i are dimensionless quantities fixed by the boundary conditions [7]. The resonance frequency ω i associated with each mode depends on α i, αi 4/α4 j = ω2 i /ω2 j. The authors have calculated the α i and the corresponding expressions of the spring constant K for each of the two modes: first mode {}}{ α =.88 and K = 0.82 kt z 2 second mode {}}{ α 2 = 4.9 and K = 0.25 kt z2 2. (3) Our spectrum analyser allows us to study the two first modes (bandwidth limitation). From figure 3 we measure the 35

4 Table. The table gives the spring constants of rectangular cantilevers measured with the thermal fluctuation method (first and second modes, K th and K th2 ) and with the method of Sader (K Sader ). K man corresponds to the spring constant indicated by the manufacturer. R to R6 are rectangular cantilevers (Thermomicroscopes, cantilever B) from the same wafer. K th K th2 K Sader K man (mn m ) (mn m ) (mn m ) (mn m ) R R R R R R Figure 6. The curve presents the ratio z2 2 / z 2 as a function of the laser spot position on a rectangular cantilever. There is very good agreement between the theoretical model [7] (dotted curve) and the experimental data. The dotted curve is the actual theoretical curve, which has also been reproduced and shifted by 9 µm to show its overlap with the experimental points (solid curve). When z2 2 =0, the curve z 2 2 / z 2 presents a minimum. This illustrates perfectly the fact that we measure the inclination dz/dx rather than the displacement, since the latter is a maximum (see the figure inset). The black squares and circles correspond to two independent experiments showing the reproducibility of the measurements. resonance frequencies associated with each mode. The ratio dz/dx is in good agreement with the values given by the theory, α 2/α2 2 = 0.6. Note that the first vibration mode contributes only 70% to z 2. According to relation (3), we can make independent measurements of the spring constant with different peaks. In figure 6 we plotted the ratio z2 2 / z 2 as a function of the laser spot position on a rectangular cantilever. We have compared it to the theoretical model (dotted curve). Again we found very good agreement between the theoretical model and the experimental data. The shift between the theoretical and experimental curves of 9 µm could be explained by air damping, by the imperfect geometry or imperfect anchorage (small variation of the cantilever inclination at point x = 0). The profile curve z2 2 / z 2 presents a minimum at the point (85 µm, z2 2 / z 2 = 0). This point corresponds to the zero difference of angles when the amplitude is maximum in the second mode (figure 6 inset). This illustrates perfectly the fact that what we measure is not the displacement but the inclination dz/dx. All these results demonstrate clearly that the model of Butt and Jaschke can be used to calibrate rectangular cantilevers with the thermal fluctuations method. We now turn our attention to the comparison between the thermal fluctuations and Sader methods [4, 5]. The method of Sader relies on the measurement of the resonance frequency, on the quality factor in air, and on the knowledge of the plan view dimensions of a rectangular cantilever. Note that the plan view dimensions have to be precisely measured for each cantilever because from cantilever to cantilever the dimensions are not conserved. Table gives the spring constant of a rectangular cantilever measured with the thermal noise method (in the case of the first and second modes) and with the method of Sader. The six cantilevers have been taken from the same wafer. As can be seen from table, excellent agreement is obtained between the two methods. In contrast, it should be noted that the cantilever spring constant varies from cantilever to cantilever. This reflects the expectation that the material properties and geometry of all cantilevers are not uniform across the entire wafer. Triangular cantilever. Often V-shaped cantilevers are used. Recently, Stark et al [26] have investigated theoretically the thermomechanical noise of a triangular cantilever by means of a finite element analysis. The modal shapes of the first ten eigenmodes are given together with the numerical constants needed for a calibration using the thermal noise method. The cantilever of type E was used in their calculation (figure 7). They found that the spring constant of this cantilever can be expressed by the following equation: K = kt z 2. Recently Neumeister and Ducker [6] have established equations involving geometry and material constants to describe the stiffness of a V-shaped AFM cantilever in three dimensions. An experimental determination of the stiffness of the cantilever using this model necessitates a precise knowledge of the geometry and above all the material constants such as the Young s modulus E. In addition, the spring constant depends strongly on the cantilever thickness t, which is difficult to measure. As in the equation, established by the authors, the stiffness of the cantilever is proportional to Et 3. We can use the Sader method to deduce the value of Et 3 from the measurements with the rectangular cantilever assuming that this value remains constant over all cantilevers (rectangular and triangular) of a single chip. The procedure involves calibration of the rectangular cantilever, from which the product Et 3, is evaluated using Et 3 = K 4L3 b. We have determined the spring constant K using the Sader Neumeister and Ducker method (SND). In table 2, we present results for the calibrated cantilevers, determined by the manufacturer and by the cited methods. In the case of cantilevers E, 36

5 Measuring the spring constant of atomic force microscope cantilevers: thermal fluctuations and other methods (a) (b) depends strongly on the cantilever thickness t, a thickness variation of 5% results in a change in the spring constant of 5%. This may explain the discrepancy, regarding the measured value of the V-shaped cantilever (type E), between the SND method and the thermal fluctuation method. Figure 7. The AFM cantilever (type E) shape with its typical dimensions. (a) The geometry used by Neumeister and Ducker [5]. (b) The real geometry of the cantilever as viewed by the optical microscope with high magnification. It is clear that if we take the real geometry of the cantilever the spring constant will be stiffer than if we take the dimension of figure (a). Table 2. The table presents the measured values of the V-shaped cantilever spring constant of type E and D from two chips of the same wafer (Thermomicroscope) using the thermal fluctuation and SND methods. K th K Neu K man (mn m ) (mn m ) (mn m ) RE RE RD RD there is clearly a discrepancy between the thermal fluctuation and SND methods. The spring constant measured using SND methods is stiffer than those measured using the thermal fluctuation model. This disagreement would have been more pronounced if we had taken into account the real geometry of the cantilever, because the measured spring constant using the SND method would be stiffer than the indicated values (see figure 7). In contrast, we found very good agreement between the two methods in the case of V-shaped cantilevers D. To investigate the disagreement regarding the lever E, we have adopted the slope method to calibrate the V-shaped E cantilever [9 2]. This method consists of mounting on the piezoelectric element a calibrated rectangular cantilever so that its free end will come into contact with the V-shaped cantilever. The deflection of the triangular cantilever is measured as a function of the piezoelectric displacement. The spring constant is determined from the slope of the force curve, using linear region. Using the calibrated rectangular cantilever, we measured the deflection of the AFM cantilever (type E). The measured spring constant value obtained is K = 90±0 mn m, which is in agreement with the measured value using the thermal fluctuation method. It should be noted that, in the case of the SND method, the assumption that Et 3 is constant over all cantilevers on a single chip could have a dramatic consequence on the evaluation of the spring constant. In fact, as the stiffness of the cantilever Conclusion The thermal fluctuation measurements of a series of AFM cantilever beams have been performed and compared to a recent theoretical model. We found a very good agreement between the experimental data and the theory of Butt and Jaschke. We have confirmed that the values of the calibrated AFM cantilevers varies from cantilever to cantilever and differs from those given by the manufacturer. In the case of the rectangular cantilever, a good agreement was found between the thermal fluctuation and Sader methods. We have shown that the thermal fluctuation method is a valid and accurate approach to the evaluation of the spring constant of V-shaped AFM cantilevers. In some cases, the calibration of V-shaped cantilevers using the SND method gives values different to those determined using thermal fluctuation. This disagreement is probably due to the non-conservation of the material properties of all cantilevers from a single chip. References [] Frisbie C D, Rozsnyai L F, Noy A, WrightonMSand Lieber C M 994 Science [2] Grandbois M, Rief M, Clausen-Shaumann H and Gaub H E 999 Science [3] Rief M, Oesterhelt F, Heymann B and Gaub H E 997 Science [4] Maaloum M, Courvoisier A 999 Macromolecules [5] Strunz T, Oroszlan K, Schäfer R and Güntherodt H-J 999 Proc. Natl Acad. Sci. USA [6] Clausen-Schaumann H, Rief M, Tolksdorf C and Gaub H E 2000 Biophys. J [7] Florin E L, Moy V T and Gaub H E 994 Science [8] Hoh J H, Hillner P E and Hansma P K 994 Proc. Microsc. Soc. Am [9] Heymann B and Grubmuller H 2000 Phys. Rev. Lett [0] Hutter J L and Bechhoefer J 993 Rev. Sci. Instrum [] Cleveland J P, Manne S, Bocek D and Hansma P K 993 Rev. Sci. Instrum [2] Li Y Q, Tao N J, Pan J, Garcia A A and Lindsay S M 993 Langmuir [3] Senden T J and Ducker W A 994 Langmuir [4] Sader J E and White L 993 J. Appl. Phys. 74 [5] Sader J E, Larson I, Mulvaney P and White L R 995 Rev. Sci. Instrum [6] Neumeister J M and Ducker W A 994 Rev. Sci. Instrum [7] Butt H-J and Jaschke M 995 Nanotechnology 6 [8] Gibson C T, Watson G S and Myhra S 996 Nanotechnology [9] Torii A, Sasaki M, Hane K and Shigeru O 996 Meas. Sci. Technol [20] Tortonese M and Kirk M 997 Proc. SPIE [2] Serre C, Pérez-Rodríguez A, Morante J R, Gorostiza P and Esteve J 999 Sensors Actuators [22] Sader J E, ChonJWMandMulvaney P 999 Rev. Sci. Instrum [23] Maeda N and Senden T J 2000 Langmuir [24] Sader J E 995 Rev. Sci. Instrum [25] Radmacher M, Cleveland J P and Hansma P K 995 Scanning 7 7 [26] Stark R W, Drobek T and Heckl W M 200 Ultramicroscopy