Quantitative analytical atomic force microscopy: a cantilever reference device for easy and accurate AFM spring-constant calibration

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1 INSTITUTE OFPHYSICS PUBLISHING Meas. Sci. Technol. 15 (2004) MEASUREMENTSCIENCE AND TECHNOLOGY PII: S (04) Quantitative analytical atomic force microscopy: a cantilever reference device for easy and accurate AFM spring-constant calibration Peter J Cumpson 1, Charles A Clifford 1 and John Hedley 2 1 Quality of Life Division (QLD), National Physical Laboratory, Teddington TW11 0LW, UK 2 Institute for Nanoscale Science and Technology (INSAT), Newcastle University, Newcastle-Upon-Tyne, UK Received 31 December 2003, in final form 15 April 2004 Published 16 June 2004 Online at stacks.iop.org/mst/15/1337 doi: / /15/7/016 Abstract Calibration of atomic force microscope (AFM) cantilevers is necessary for the measurement of nanonewton and piconewton forces, which are critical to analytical applications of AFM in the analysis of polymer surfaces, biological structures and organic molecules. We have developed a compact and easy-to-use reference artefact for this calibration by bulk micromachining of silicon, which we call a cantilever microfabricated array of reference springs (C-MARS). Two separate reference cantilever structures, each nominally 3 µm thick, are fabricated from a single crystal silicon membrane. A binary code of surface oxide squares (easily visible in light, electron and atomic force microscopy) makes it easy to locate the position of the AFM tip along the length of the cantilevers. Uncertainty in location is the main source of error when calibrating an AFM using reference cantilevers, especially for those having spring constants greater than around 10 N m 1. This error is effectively eliminated in our new design. The C-MARS device spans the range of spring constants from 25 N m 1 down to 0.03 N m 1 important in AFM, allowing almost any contact-mode AFM cantilever to be calibrated easily and rapidly. Keywords: atomic force microscopy, calibration, nanonewton/piconewton force, measurement by AFM 1. Introduction Reference artefacts for dimensional calibration [1] of AFM have been available from many sources for ten years or more. The determination of the shape of the tip also has been the focus of a number of calibration methods [2 4]. Moving beyond dimensional measurements, there is an increasing need for the accurate measurement of forces between 10 pn and 10 nn by AFM, particularly for surface analytical purposes such as the analysis of polymer surfaces and biomolecules [5], and for this an accurate knowledge of the cantilever spring constant is needed. However, the calibration of the spring constant of AFM cantilevers is more subtle than calibrating the dimensions for the x,y and z axes [6]. Uncalibrated AFM cantilevers can lead to very large errors in the measurement of nanonewton forces [7], such as in direct experiments to break individual covalent bonds by AFM. One reason why similar cantilevers have very different spring constants is that the thickness of AFM cantilevers is difficult to control to the exquisite tolerance required to give them repeatable spring constants. An easy and accurate method of calibrating cantilevers is a crucial need within the nanotechnology community. A number of methods have been suggested [8 24] for the calibration of AFM cantilevers, and recently reviewed [24, 25]. Particularly effective is the use of a reference /04/ $ IOP Publishing Ltd Printed in the UK 1337

2 P J Cumpson et al cantilever [15], in which the spring constant of the AFM cantilever is measured by comparing it with the known spring constant of a reference cantilever. This is often known as the cantilever-on-cantilever method, but of course relies on the user having a reference cantilever that is easy to use and of known spring constant. In this work, we describe the design of a new reference cantilever for use in the cantilever-on-cantilever procedure. It has a number of advantages in terms of ease-of-use. The principle advantage of the design described here is a built-in length scale visible in AFM images that essentially eliminates the greatest source of uncertainty in the method. 2. Methods and materials The devices are made by bulk micromachining of silicon. A 3 µm single crystal membrane was created by bulk backside etching of silicon in a foundry process similar to that used in the manufacture of membrane pressure sensors. An electrochemical etch-stop [26] method was used to ensure that a uniform and low-roughness membrane was created. Cantilever structures were created by cutting slits in the membrane by deep reactive ion etching (DRIE) to ensure that the sidewalls of the cantilevers are as near perpendicular to the surface as possible. Key features of the design are as follows: (1) A cantilever and a simply-supported beam are fabricated on the same chip, catering for small and large AFM spring constants respectively. (2) The cantilever and the beam are 150 µm in width and 1.6 mm in length, allowing AFM users to approach and land tips easily, even given the imprecision of some grossapproach stepper motor systems. (3) A binary ruler stretching the length of the cantilevers allows accurate placement of an AFM tip at a point along the middle of the cantilever. This makes it straightforward to determine the exact position of the tip from an AFM image. The ruler provides a linear array of 26 points at which the spring constant is known accurately. (4) A glass substrate is anodically bonded to the underside of the wafer for easy handling by the AFM user. (5) Surface piezoresistors near the base of the cantilever allow the monitoring of displacement and vibrations of the cantilever [27], if required. Optical micrographs of the finished die are shown in figure 1. A schematic vertical cross-section through the length of the cantilever is shown in figure 2. The long axis of the cantilever is aligned with a 110 crystallographic direction of the silicon, so that, for deflection of the cantilever, the appropriate literature value for Young s modulus is 168 GPa [26]. Particular care was taken to select a fabrication process that gives a uniform cantilever thickness. In bulk micromachining, the implantation of an etch-stop layer (for example boron or oxygen) often fails to produce wellcontrolled thin membranes of the type required [26]. We used a process incorporating an electrochemical etch-stop method to produce uniform and low-roughness undersides for the reference cantilevers. In brief, a lightly doped p n junction was used as an etch stop by applying a bias between the wafer and the counter electrode in the etchant. This method Figure 1. Optical micrographs of the cantilever microfabricated array of reference springs (C-MARS). Note the binary length scale formed from 10 µm surface oxide squares. is known to give thickness uniformity of better than 1% in 10 µm membranes irrespective of etch uniformity or wafertaper [26]. The critical factor regarding thickness uniformity is the uniformity of dopant density in the first few micrometres of depth into the p-doped wafer; which over the 1.6 mm length of the reference cantilever is likely to be extremely uniform. 3. Integral length scale The main source of uncertainty in the use of a reference cantilever for AFM calibration lies in the measurement of the displacement of the AFM tip along the length of this reference cantilever. Often the AFM user does not have an optical microscope positioned to allow this measurement to be made accurately. Sometimes AFM is used with no optical microscope at all. Many designs of AFM cantilever make it difficult to estimate, from a top-view alone, the exact location of the tip below. Therefore a length scale was incorporated into the surface of the cantilever. This had the following three aims: (1) To allow easy measurement of the position of an AFM tip from a local AFM image of the surface alone. (2) That the scale should not change the mechanical properties of the cantilever significantly. (3) That the scale should define the centre-line of the reference cantilever. This allows one to move the AFM tip to the middle of the reference cantilever in order to avoid errors which would arise from the induced twisting of the reference cantilever. One solution would be to etch scale markers in the cantilever surface resembling the scale of a ruler. In fact, we do not have enough lithographic resolution within the AFM field of view to allow such a scale. Typically, the largest scan size available when using commercial AFM tube scanners is 1338

3 Quantitative analytical atomic force microscopy Figure 2. Schematic cross-section through the length of the C-MARS device. Figure 3. A 5-bit binary length scale code incorporated in the oxide layer on the top surface of the cantilever. 100 µm 100 µm, so the entire information required to deduce the exact position of the AFM tip on the C-MARS device must be available from an image of this size or smaller. An oxide layer on the upper surface was available for patterning into a length scale, but only at relatively low resolution with a minimum feature size of 10 µm 10 µm. We devised a binary length scale that contains sufficient information within the field of view of an AFM, and yet requires only 10 µm 10 µm squares. Figure 3 shows an example of this binary code. The code can be read very easily by AFM (shown in figure 4), optical microscopy (shown in figure 1) and scanning electron microscopy (shown in figure 5). The scale consists of simple arrows made up of 10 µm 10 µm squares and separated by 60 µm from the next arrow. Between successive arrows is a 5-bit binary code defining the position for the arrow preceding it. The length scale has a secondary purpose in defining for the AFM user a line running down the middle of the cantilever; it is known [28, 29] that deviation from this middle line when calibrating an AFM cantilever against a reference cantilever can lead to errors arising from the torsional strain component induced in the reference cantilever. Figure 4. AFM image showing length scale on the surface of the C-MARS device. The clarity of images such as this allows easy positioning of the AFM tip to an accuracy of around 500 nm prior to acquiring a force distance curve. This virtally eliminates the main source of error in the calibration of AFM cantilevers by the cantilever-on-cantilever method. Figure 4 shows one of the binary length scales imaged by AFM. One concern prior to manufacture of the C-MARS devices was that the large size and consequently low resonant frequency of the reference cantilever would result in excitation during imaging by AFM, making such images very noisy. In fact, despite the relatively low fundamental resonant frequency of the reference cantilever (around 1.41 khz) we experienced no difficulty in imaging the scale. In fact, even close to the end of the reference cantilever there was no obvious degradation in image quality. Figure 5 is a montage of scanning electron microscope (SEM) images of the C-MARS device. In figure 5(a) we have pulled the cantilever into contact with the glass substrate electrostatically to allow the imaging of the edge of the membrane from which it is formed. This edge shows ripples characteristic of the deep reactive-ion etching (DRIE) process that formed it. Figure 5(b) shows the underside of the reference cantilever. A patch of conductive 1339

4 P J Cumpson et al Figure 6. Surface topography on the top and underside of the reference cantilever, determined by white-light interferometry. Each of the two images is 100 µm 200 µm micrometres in size. Root-mean-square roughnesses are 0.9 nm and 6.5 nm for the top and underside of the cantilever respectively Sader s method Figure 5. Scanning electron microscope images of the reference cantilever surface, using a primary energy of 20 kev. (a)thetop surface of the specimen, with the cantilever pulled downwards to reveal the edge of the silicon membrane from which it is made. (b) The underside of a cantilever removed by pressing carbon-loaded adhesive tape onto the specimen surface and pulling the entire cantilever away with it; this allows us to look at the reverse side of the cantilever and hence the quality and uniformity of the etch-stop surface. This is excellent. tape was first attached to the surface of the reference cantilever die, and then pulled off, taking the cantilever and some surrounding part of the membrane with it. The final SEM image in figure 5(b) confirms the quality of the electrochemical etch stop in giving rise to a smooth and geometrically precise surface on the underside of the cantilever. Figure 6 shows topographic images of 200 µm 100 µm areas from the top of the die (beside the cantilever, and therefore having a similar surface roughness) and the underside of the cantilever (the surface that was bulk etched). Surface roughnesses are 0.9 nm and 6.5 nm respectively, well below the level that we would expect to lead to uncertainty in the comparison of spring constants. These topographic images were acquired by white-light interferometry. 4. Determining the spring constant of the reference cantilever We now compare three different methods for determining the spring constant at the end of the C-MARS device we have constructed. Sader et al [14] showed that for a diving-board cantilever of sufficiently large length-to-breadth ratio (a condition which our reference cantilever fulfils), the spring constant at the end of the cantilever, k E,isgivenby k E = 4π 2 κ Sader mf 2 (1) where κ Sader = , m is the mass of the cantilever and f is its fundamental resonant frequency. Sader et al [14] considered corrections for measurement of the resonance in air, but ours were measured in vacuum by Doppler velocimetry and needed no correction. We can estimate the mass of the cantilever using its linear dimensions and a reference value for the density of silicon of 2330 kg m Euler Bernoulli theory For a clamped cantilever undergoing small and therefore linear displacements, the spring constant at the end is given in terms of Young s modulus, E, of the material by k E = Ebt3 (2) 4L 3 where t is the thickness of the cantilever, b is the breadth of the cantilever, and L is the length of the cantilever Finite element analysis Neither of these methods properly accounts for the boundary conditions at the base of our cantilever, because the cantilever is not clamped but cut within a membrane of the same 3 µm thickness. This is difficult to account for using analytical models giving equations in closed form, so instead we performed finite element calculations using the ABAQUS Finite Element code within the Coventorware 2003 package 3. The values for k E calculated by these three methods for our reference cantilever are presented in table 1. The Euler Bernoulli method and finite element method both suggest a higher value for the spring constant at the 3 Coventor, Inc., 625 Mount Auburn St., Cambridge, MA 02138, USA. 1340

5 Quantitative analytical atomic force microscopy Figure 7. The interferometer/vibrometer instrument used in measuring resonant frequencies and mode shapes of the C-MARS device. Table 1. Spring constant at the end of the C-MARS device, estimated by three methods. Agreement is good. Method Sader et al [14] Euler Bernoulli Finite elements Defining k E = 4π 2 κ Sader mf 2 k E = Ebt3 Constitutive 4L equation 3 equations Value N m N m N m 1 end of the cantilever. This may be, for example, due to our membrane being around 8% thinner than the nominal value of 3 µm. We will use the Sader method value of k E = N m 1, because the resonance frequency measurement it uses renders the result only linearly dependent on the thickness of the cantilever. Each C-MARS device must be individually calibrated by the manufacturer or a calibration laboratory before the AFM user receives it, whether by the Sader method or some other. If the Sader method is used then this involves measurement of its fundamental resonant frequency; this can be conveniently performed in vacuum by sweeping the frequency of a vertical piezo-vibrator while monitoring the vibration amplitude using the built-in piezoresistors. This is a rapid electrical measurement. 5. Resonance measurements The resonant frequencies of the cantilever can be measured using built-in piezoresistors as displacement sensors. However, it is useful to plot not simply the frequencies, but also the modes of vibration. The value of Sader s factor κ Sader is valid only for the fundamental mode, so we must be completely confident that it is the fundamental mode that we are observing. We identified the modes of vibration of the cantilever using a Doppler laser vibrometer system as previously described [30]. The cantilever chip was vibrated sinusoidally using a piezoelectric actuator, and the resonant frequencies of the cantilever determined, as shown schematically in figure 7. The fundamental resonance peak is shown in figure 8 for two different background pressures. One can see that the sharpness of the resonance increases substantially as the pressure decreases. Even at 10 Pa pressure, the large area of the cantilever means that damping is high and the quality factor of the resonance rather low for a MEMS device. This allows AFM imaging of the surface of the cantilever in air with no degradation of image quality, even though the cantilever has a low resonant frequency. Conversely, the measurement of its resonant frequencies (for example, the fundamental frequency that is used in Sader s method of estimating cantilever spring constant) must be performed in vacuum, at a pressure below about 5 Pa, whether using the built-in piezoresistive sensors or the Doppler velocimetry technique. Figure 9 shows the shape of the fundamental mode of the cantilever, again measured by Doppler velocimetry. 6. Calibration procedure So far we have described the design and manufacture of the C-MARS device, and how it can be calibrated by the manufacturer or calibration laboratory. The small mass of the cantilever means that it is relatively robust to mechanical shock and can easily be sent to the AFM user by mail. Now we shall discuss how the user can measure the spring constant of an arbitrary AFM cantilever by means of this device. A simple two-point calibration is rapid and effective for the busy AFM user wishing to calibrate an AFM cantilever. Two-point calibration uses the ratio of the slopes of two force distance curves one pressing the AFM tip against a stiff surface, one against the reference cantilever to obtain 1341

6 P J Cumpson et al Figure 8. Fundamental resonance of the reference cantilever, measured at pressures of 3 Pa and 10 Pa. The ordinate is proportional to the velocity amplitude of vibration near the end of the cantilever, and is in arbitrary units. Table 2. Nominal spring constant at indicated positions along the length of the reference cantilever. Figure 9. Experimentally measured fundamental mode of vibration of the reference cantilever at 1.41 khz. The fundamental mode corresponds to the resonant peak shown in figure 8, and is shown using full circles. A continuous line is plotted to guide the eye as to the approximate shape of the mode, but is not intended to represent the mode shape with numerical accuracy. In particular, measurements do not extend the full 1.6 mm length of the cantilever. a measurement of the AFM cantilever spring constant with an uncertainty in the range of ±5% to ±10%. The largest contributor to this uncertainty is typically stick-slip processes that take place at the contact between AFM tip and reference cantilever, so there is some scope in future to reduce this uncertainty by, for example, depositing a layer having appropriate tribological properties on the top surface of the reference cantilever. The procedure for two-point calibration of an AFM cantilever using the C-MARS is as follows: (1) Find the nominal spring constant in table 2 closest to the specification of the AFM cantilever that you wish to calibrate. Note the binary code corresponding to this nominal spring constant. (2) Move the AFM tip to the section of the C-MARS reference cantilever showing the binary code illustrated in figure 10, and within 20 µm of the middle line of the cantilever. The Nominal spring Distance Distance constant from base of from end of k ref cantilever, cantilever, Binary Code, (N m 1 ) L (µm) (L 0 L) (µm) code, l l two central columns of table 2 show the distance of this point from the base and the end of the cantilever, which may help locate this point rapidly. If you have an optical microscope it may be simpler just to move the microscope stage until you see this binary code. (3) Acquire the first force distance curve of Z-range not more than 500 nm. (4) Move the tip of the AFM cantilever to a thick part of the die, away from the reference cantilever. Acquire the second force distance curve. 1342

7 Quantitative analytical atomic force microscopy Figure 10. Optimum position for force distance curve acquisition for calibration of AFM cantilevers. The exact position of the base of this arrow can be determined from the 5-bit binary code that follows it, as shown in figure 3. Consider the second force distance curve acquired from the stiff surface, such as the neighbouring area of the chip consisting of silicon 0.4 mm thick. Since this surface is rigid, the increased deflection of the tip during acquisition of a force distance curve is equal to the downward displacement ( Z) of the cantilever by the tube scanner of the AFM, i.e., α V Stiff = Z Stiff (3) where α is a factor depending on the design of the cantilever under test, and the laser spot position on this AFM cantilever. V Stiff is the voltage signal (in some designs of AFM instruments a current signal) originating in a split photodiode, which is linearly related to the deflection of the cantilever under test. Now consider the first force distance curve from the region of the reference cantilever having a spring constant k ref close to the expected spring constant of the AFM cantilever being calibrated. The equation for this case is k ref α V CMARS = k ref + k c cos θ Z CMARS (4) where θ is the angle of the cantilever with respect to the surface, and k c is the spring constant of the cantilever under test. For all commercial AFM instruments θ is rather small, typically around 4 to 15, so that the cosine factor is not very significant, and one does not need to know θ to an accuracy of better than a few degrees to obtain an accurate spring-constant calibration. We can eliminate α and calculate the spring constant of the cantilever from [ ( VStiff ) ] k c Z = ( Stiff k VCMARS ) 1 sec θ ref Z CMARS (5) where, for the cantilever whose measurements are plotted in figures 8 and 9, k E = ± N m 1. The relationship between k ref and k E will now be described. 7. Deviations from the theory for an Euler Bernoulli clamped beam For a Euler Bernoulli clamped cantilever, ( ) 3 L0 k ref = k E (6) L Figure 11. Schematic top view of the cantilever, showing the effective base of the cantilever, used in the text to develop a semi-empirical expression, equation (9), for cantilever-on-cantilever calibrations using this cantilever cut from within a membrane. where L 0 is the length of the reference cantilever and L is the distance from the base of the reference cantilever to the point at which k ref is measured. However, this is an inadequate description of the C-MARS cantilever because it is not firmly clamped, for reasons we now describe. To provide an effective length scale on the surface of the C-MARS device, the origin of the scale must be precisely aligned with the base of the cantilever. A common problem in bulk micromachining is the difficulty in aligning an etch mask on the back of the wafer with a mask on the front; the technology of lithography, inherited from the microelectronics industry, focuses instead on the precise alignment of multiple front-side masks. This makes it very difficult to make a clamped cantilever with a length scale on its top surface precisely aligned with the reference cantilever base. Instead, we used a front-side DRIE etch through a clamped membrane formed from etching from the back side, and ensured that the DRIE etch from the front was precisely aligned to the oxide length scale, also patterned from the front. This gave cantilevers with good, reproducible length scales, but meant that we cannot use the simple Euler Bernoulli equation without modification to account for the fact that the base of the cantilever is attached to a membrane and not firmly clamped. Analytical solutions for such complex geometries in elastic plates are generally not available. Instead, we used finite element analysis (the ABAQUS code within the Coventorware 2003 package (see footnote 3)) to calculate the spring constant at a large number of points along the reference cantilever and fitted this using a semi-empirical generalization of the Euler Bernoulli model. This semi-empirical model aims, as illustrated schematically in figure 11, to account for some of the additional flexibility arising from the membrane by (1) modelling the reference cantilever as an Euler Bernoulli clamped cantilever, longer by L than its true geometrical length, and (2) modelling it as clamped to an additional spring of spring constant k base at its base which is, in turn, rigidly clamped to the die. The first accounts for our intuitive expectation that there is some flexure in the cantilever close to the base where one would expect almost none if it were rigidly clamped. The second accounts for some of the compliance of the membrane 1343

8 P J Cumpson et al Deviation from FEA result 10% 5% 0% -5% -10% kref (N/m) Figure 12. Deviation of semi-empirical model, equation (7), from finite element analysis (FEA) results. This deviation is less than ±1% for all spring constants below about 25 N m 1. itself. These two new parameters were obtained by leastsquares fitting to finite element calculation results. This results in an expression for the spring constant of the cantilever under test, 1 k c cos θ = 1 k base ( )( ) [ ( L + L 3 VStiff ) ] 1 Z ( Stiff k E k base L 0 + L VCMARS ) 1 (7) Z CMARS where k base represents the spring constant at the base of the cantilever, at the point indicated in figure 11. We can substitute for the length ratio in terms of the index l that appears on the corresponding binary code on the cantilever surface. l is the index number of the position of the AFM tip on the C- MARS reference cantilever (i.e. as encoded on the surface in binary oxide squares). The first such code, l = 1, appears 32 µm from the base of the cantilever, and they are spaced 60 µm apart, so that if all length variables are expressed in micrometres, L 0 + L L + L = hence 1 k c cos θ = 1 ( 1 + k base ( 60l +32+ L L 60l +32+ L 1 ) k E k base ) [ ( 3 VStiff ) ] 1 Z ( Stiff L VCMARS ) 1 (9) Z CMARS where L = 43.7 µm, k base = 51.1 Nm 1,andk E is a constant for a given cantilever. Figure 12 shows the fractional difference between the prediction of equation (9) and finite element calculations. This is less than 1% for all local spring constants on the cantilever below around 25 N m 1, which is sufficient given the uncertainties of the cantileveron-cantilever calibration method and the uncertainty of the finite element calculations themselves. Given equation (9), reference cantilevers distributed by a calibration body or National Measurement Institute may, for example, be accompanied by a calibration certificate giving the value of k E for that particular cantilever, for example by Sader s method of equation (1), or another validated method. Work in future will focus on methods of calibrating C-MARS devices to the SI, along the lines of the electrical nanobalance [31] or other traceable methods [32]. (8) k c (N/m) Location of AFM tip along reference cantilever (µm) Figure 13. Experimental AFM force distance measurements for an AFM cantilever of nominal spring constant 0.5 N m 1. Each point represents the average of between four and six force distance approach-curve slopes. The broken line represents the uncertainty-weighted value of N m Atomic force microscopy measurements To check the validity of this procedure we performed a number of two-point calibrations on a typical AFM cantilever; this is more effort than can be spared by the typical AFM user, but plotting a number of data points will add to our confidence in the calibration method. Figure 13 shows a plot for a cantilever of nominal spring constant 0.5 N m 1. This calibration indicates the true spring constant of this AFM cantilever to be ± N m 1. Figure 14 presents finite element analysis results addressing the question of how close the AFM tip must be to the middle of the reference cantilever to ensure that the error due to cantilever twisting is negligible. Within 20 µm of the mid-line of the cantilever this error is less than ±0.5%, which is negligible in comparison with the other uncertainties of the cantilever-on-cantilever method. 9. The effect of damping in air and liquids Figure 8 gives convincing evidence that at atmospheric pressure (around 10 5 Pa) we should expect the resonance of such a large reference cantilever to be heavily damped. This is a major advantage, since it allows us to (a) image the built-in length scale and (b) acquire force distance curves free of oscillations. If not damped, the cantilever would be easily excited into resonance. However, the same air damping might be expected to lead to a maximum rate at which force distance curves can be acquired without the reference cantilever appearing to be stiffened by fluid viscous damping. One would expect an even greater effect in liquids. First recognize that this drag force will be a constant force on the cantilever while being depressed at constant speed. Therefore, since the calibration requires only measurement of the slope of a force distance curve, to first order the effect of viscous forces is zero. Nevertheless, since our C-MARS cantilever is larger than those used previously, we should estimate the magnitude of this viscous damping force. 1344

9 Quantitative analytical atomic force microscopy Figure 14. Contour plot showing, in units of per cent deviation from the spring constant in the middle line of the reference cantilever, the error due to cantilever twisting. This was calculated by finite element analysis. This error is below 0.5% within 20 µm of the middle line of the cantilever. The motion of the reference cantilever under the force applied by the AFM tip occurs at very low Reynolds number at atmospheric pressure, i.e., Re = ρvd (10) η where Re is Reynolds number, v is the velocity of the cantilever, ρ 1.24 kg m 3 is the density of air, D = 150 µm is the characteristic dimension of the reference cantilever (its width) and η = Nm 2 s is the dynamic viscosity of air. The velocity of the cantilever, v, typically corresponds to a displacement of the order of 100 nm over an interval of 1 s or so during acquisition of a force distance curve, so that v ms 1. This suggests a Reynolds number of around 10 6, meaning that the air resistance on the cantilever is almost completely viscous. Therefore we can estimate the viscous drag on the cantilever in a gross fashion, modelling the cantilever as a series of discs with similar cross-sectional area, F drag = 16nηDv (11) where n L/b 11, so that F drag 0.05 pn in air (η = Nm 2 s) and F drag 2pNina typical aqueous solution (η = Nm 2 sat30 C). This is negligible compared to the reaction force being measured. The reference cantilever can therefore be used with confidence. 10. Conclusions We have designed and tested a new reference cantilever, which we call C-MARS, for the rapid and accurate calibration of AFM cantilever spring constants. This has a number of advantages, as follows: (a) The built-in length scale in this new reference cantilever design eliminates the main source of uncertainty in AFM spring-constant calibration by the cantilever-oncantilever method. (b) The large surface area of this reference cantilever, relative to the accuracy of the gross-approach mechanism of most AFM stages, makes landing the AFM tip on the reference cantilever easy. (c) The reference cantilever incorporates piezoresistors designed to allow resonant frequency to be measured, and as a potential independent measure of reference cantilever deflection. (d) Errors due to partial twisting of the cantilever are avoided by moving the AFM tip to the mid-line of the length scale after contact. The error due to reference cantilever twisting can be made negligible by acquiring force distance curves within 20 µm of the mid-line of this cantilever. (e) The relatively large area of the reference cantilever means that its resonant frequency is low (around 1.4 khz). Nevertheless, AFM images of the length scale on the cantilever are free of noise arising from the excitation of the reference cantilever due to effective air damping, again due to the relatively large surface area of the reference cantilever. (f) The new reference cantilever allows a precision of around ±3% in the calibration of AFM cantilever spring constants, and accuracy when expressed in SI units of around ±10% at the one-standard-deviation level, though additional work is needed to make the device traceable to the SI. (g) The spring constant of AFM cantilevers can be calculated quickly and easily from equation (9). Future research will focus on methods of calibration of C-MARS transfer standards so as to make them traceable to the SI. Acknowledgments The authors wish to thank Tony Rogers (AML Ltd, UK) for detailed device layout assistance, Dr Joost van Kuljik (Coventor, NL) and Peter Lovelock (NPL) for finite element software advice, and Jens Ebbecke (NPL) for skilful wirebonding. Thanks are due to David Mendels (NPL) and Steve Spencer (NPL) for assistance with scanning electron microscopy. Special thanks are due to Dr Martin Seah for critical readings of the manuscript and many useful suggestions. Microfabrication was funded under NPL Strategic Research Project 9SRPQ170. This work forms part of the UK Department of Trade and Industry Valid Analytical Measurement (VAM) programme. 1345

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