Atomic Force Microscope Cantilever Flexural Stiffness Calibration: Toward a Standard Traceable Method.

The evolution of the atomic force microscope into a useful tool for measuring mechanical properties of surfaces at the nanoscale has spurred the need for more precise and accurate methods for calibrating the spring constants of test cantilevers. Groups within international standards organizations such as the International Organization for Standardization and the Versailles Project on Advanced Materials and Standards (VAMAS) are conducting studies to determine which methods are best suited for these calibrations and to try to improve the reproducibility and accuracy of these measurements among different laboratories. This paper expands on a recent mini round robin within VAMAS Technical Working Area 29 to measure the spring constant of a single batch of triangular silicon nitride cantilevers sent to three international collaborators. Calibration techniques included reference cantilever, added mass, and two forms of thermal methods. Results are compared to measurements traceable to the International System of Units provided by an electrostatic force balance. A series of guidelines are also discussed for procedures that can improve the running of round robins in atomic force microscopy.

1. Introduction

Atomic force microscopy (AFM) has been struggling with issues of calibration and accuracy since its inception in 1986 [1]. Initially, the emphasis was on dimensional accuracy since imaging was the first focus of AFM. The nonlinearity inherent in the piezoelectric ceramics used to drive most AFM scanners was addressed by either software corrections or incorporation of closed loop sensors. More recently, interest in using AFM to measure nano-scale forces has prompted researchers to deal with AFM cantilever spring constant calibration issues. This has also led manufacturers to develop better cantilevers with tighter production tolerances (and hence a smaller spread in spring constants) and researchers to develop better methods to calibrate cantilevers in the field.

As spring constant calibration techniques have proliferated, attempts have been made to compare techniques to determine which ones have the best precision and accuracy. Efforts have also begun in standards organizations such as the Versailles project on Advanced Materials and Standards (VAMAS), and the International Organization for Standardization (ISO) to understand which techniques are useful for comparing data from different laboratories around the world. One of the Authors (RG) is currently Chairman of VAMAS Technical Working Area 29 on Nanomechanics Applied to Scanning Probe Microscopy. Recently a mini round robin (MRR) was conducted among three national laboratories worldwide in an attempt to provide a foundation for a larger round robin on comparison of flexural stiffness calibration methods for AFM. This MRR had several key findings that were useful in streamlining a larger round robin (currently underway) and minimizing the possibility of damage to the samples during shipping and handling. A summary report is available online [2].

The purpose of this paper is to expand on the details of the experimental measurements that were made during the MRR and also provide additional data on the same cantilevers using techniques that were not available at the time of the MRR in order to establish the potential accuracy of the techniques.

2. VAMAS TWA 29 Mini Round Robin

The MRR was conducted among three laboratories in three countries to evaluate handling and testing protocols for determining the flexural spring constants of AFM cantilevers. The laboratories were National Laboratories in the United States (The National Institute of Standards and Technology—NIST), the United Kingdom (The National Physical Laboratory—NPL), and Japan (The National Institute for Material Science—NIMS) and included researchers who were very familiar with AFM. The study was intended as an initial foray into cantilever calibration in order to experience logistical, handling, and testing issues that might come up in a larger round robin with many different participants. By experiencing and addressing problems in the MRR it was anticipated that a future round robin could be conducted with fewer problems.

A kit consisting of six similar commercial test cantilevers from a single production batch (silicon nitride, triangular) and a commercial reference cantilever artifact, was mailed to each laboratory in sequence. A detailed description of these cantilevers is provided in Appendix A. Each laboratory was asked to perform cantilever spring constant calibration procedures on the test cantilevers using procedures with which they were familiar. Drafts of very detailed procedures for an added mass method and a reference cantilever method were written by two of the authors (RG & MR) and were included with the test kit and are also attached to this paper as Appendices B and C. These procedures included explicit instruction on how to calculate the spring constant in each case and report the values. The results of the MRR were collected and the data compared.

All three laboratories performed the reference cantilever method and the statistical analysis of the results of each laboratory indicated good agreement between the results obtained from the three laboratories. One of the laboratories (NIST) also conducted added mass calibrations, and the spring constant values obtained were consistent with those obtained with the reference cantilever method. These results are described in more detail below.

3. Reference Cantilever Method

The reference cantilever method is a straightforward method for obtaining a spring constant for an unknown, test, cantilever by performing a force curve on a known spring. By measuring the deflection of the known—unknown cantilever couple, the spring constant of the unknown can be calculated. This technique was originally popularized by Tortonese & Kirk [3] who produced some of the first microfabricated reference cantilevers used specifically for this purpose. In practice in an AFM, the technique actually requires two force curves. One on an extremely stiff surface approximated as rigid essentially the z piezo displacement effect on the laser spot translation across the photodiode (the so-called optical lever sensitivity) while the other force curve on the end of the reference spring (the reference cantilever) provides the relationship between displacement and laser spot translation for the springs in series. The defining equation used to estimate the spring constant of the unknown (test) cantilever is:

Srigid and Scant are the slopes of the compliance curves during contact with either the “rigid” (a very stiff piece of Si) or reference cantilever surfaces and typically have units of V/nm. The cosine squared correction is needed to correct geometrically for the inclined angle (φ) of the test cantilever in the AFM holder. In the case of an 11° incline, this works out to be about a 4 % correction. Note that for cantilevers that have very long tips relative to their lengths the geometric correction becomes more complex and the torque of the tip must also be taken into account [4]. For the case of the MRR, the test cantilevers used have short tips (3 μm) and they are relatively long (115 μm) so this effect is at the subpercent level and can be ignored. The second half of equation 1 represents the “off end correction” factor that must be applied. Spring constants for reference cantilevers are usually specified at the very end of the cantilever and it is not possible to actually contact there. By pressing at a specific and known location, the actual stiffness at that location can be used based on the Euler-Bernoulli model for a rectangular cross section cantilever which varies as the cube of the length.

The spring constant calculated in the reference cantilever method above is the “intrinsic” spring constant—i.e., perpendicular to the long axis of the test cantilever. As such, it is portable and can be used in any AFM instrument by just dividing by the cos2 φ of the inclined angle to give the vertical (“effective”) component of the spring constant for that particular instrument.

One key aspect of the reference cantilever procedure is the care needed in defining the precise point of contact along the reference cantilever. This feature was considered essential because the spring constant for a rectangular reference cantilever varies as the cube of the length. Small errors in placement of the contact point can therefore have large effects in measured stiffness (precision and accuracy). One approach, used successfully over the years in our laboratory uses a known length (the tip set back length, ΔLtip) as a visual internal standard to position the contact point. This alignment procedure, provided in Appendix B, was also provided to the MRR participants. The reference cantilever procedure provided in Appendix B did not place any constraints on which force curve (approach or retract) was to be used for the slope estimation. It was suggested that force curve ramp length start at 500 nm, but it could be adjusted to suit the requirements of the particular instrument. A minimum of six measurement pairs (on a “rigid” surface and on the reference cantilever) were requested from each participant to determine the statistical repeatability of the measurement from each laboratory1.

The main drawback in the reference cantilever method is that it requires the AFM tip to actually touch the surface during use. This can potentially cause damage to the tip that may affect future use. Researchers often get around this by performing the calibration after the more delicate imaging and measurements have been conducted. A second caution is that the reference cantilever calibration value is only as accurate as the reference cantilever itself. Variations in reference cantilevers as great as 30 % have been observed [5, 6] so care must be taken to ensure that accurate values are used. One significant advantage of the reference cantilever method is that it has the potential to be traceable to the International System of Units (Système International d’Unités or SI). Work is also currently underway at NIST to microfabricate a large batch of reference cantilevers (NIST SRM 3461) that would be very uniform and statistically linked to (SI) traceable measurements to reduce the current accuracy uncertainty in reference cantilevers.

4. Initial VAMAS Data Comparison

As all three labs utilized the reference cantilever method, the collated data can be easily compared and are summarized in Fig. 1. The error bars in each sample represent the standard deviation of the mean using the six repeat data on each specific cantilever. The three labs used different AFM instrumentation. Lab “A” used a Veeco2 Dimension 3100, lab “B” used a Veeco Multimode, and lab “C” used a Park XE100 system.

Fig. 1

Comparison of reference cantilever calibration results from three different laboratories.

The average relative uncertainty of the measurements for all six cantilevers was 11 % (for lab “A”) and 6 % (for labs “B” and “C”). Three things are immediately apparent from the graph. First, all three labs had statistically similar results and therefore the reproducibility of the prescribed method from lab to lab was very good. Second, the six test cantilevers were similar in spring constant suggesting that for these particular cantilevers selecting a subset of cantilevers within a single manufacturing batch (based on resonance frequency for example) can be fairly effective at reducing the cantilever-to-cantilever variation. Third, the spring constants estimated from the reference cantilever method were all lower than the nominal value assigned by the manufacturer by about 30%.

While the average repeatability of the measurements for lab “A” was typical for this type of measurement reported in the literature [7] (± 10 % to ± 30 %), the observation of the variation of error bars for lab “A” suggests a discreteness in the data that points to a statistical analysis issue. Test #1 had a relative uncertainty of ± 17 % while test #3 had no uncertainty. Looking at the raw data indicated that the problem lay with the low number of reported significant digits (two) from that laboratory. Since the actual numbers reported for the slope of the force curve measurement in Eq. (1) (e.g. 0.014 V/nm) had a small first digit, change of just one digit in the second number is a change of almost 10 %. The effect is to blow up or nullify small variations in the data, depending on where the data lay relative to the last significant digit and explains the discrete nature of the data. This highlights the need to specify a minimum number of significant figures (in this case three) for the raw data in the round robin.

The procedure for this MRR did not specify whether to use the approach or retract portion of the force curves to calculate the slope of the compliance curves. In retrospect, that was a dangerous omission that could have affected the comparison of data from different laboratories. It was fortunate that the selected test cantilever had a short (nominally 3 μm) tip that had retract curves that were very similar in slope to approach curves so it didn’t matter, but that will not always be the case. Pratt et al. [8] showed that in some circumstances, approach and retract slopes can be very different and the spring constants calculated from them will vary. They attributed the effect to friction between the tip and surface that gets amplified by the geometric leverage of tip height and causes hysteresis in the force curves. They recommended an average of both curve slopes be used to reduce the influence of the effect on the estimated spring constant.

The absolute value of the spring constant of the test cantilever obtained in the reference cantilever method is based on the value for the reference cantilever; therefore, the accuracy of the method depends on the accuracy of the reference cantilever. In the case of this study, that absolute value was based on the manufacturer’s nominal spring constant value which was given as “0.711 N/m” for all five “long” reference cantilevers in the set. For the purposes of comparing test results among the three laboratories, the actual absolute value did not matter since all participants used the exact same artifact (the long reference cantilever). Essentially, the comparison provides the relative precisions of the calibrations and how they might be biased by the instruments themselves.

The reference cantilever used in the MRR was selected from a batch of five cantilever sets purchased from the manufacturer (CLCF-NOBO, Veeco Probes, Camarrillo, CA) and utilized the long reference cantilever with a resonance frequency closest to the nominal value specified in the hope that the nominal spring constant would represent a more accurate value. An estimate of the spring constant of the reference cantilever, performed using the Sader method [9] using the web-based Java applet [10] indicated a stiffness of 0.70 N/m, which was close to the nominal value. This suggested that the manufacturer-assigned value for the MRR was reasonable for that particular cantilever. Another long reference cantilever from the same purchased set had a Sader-estimated [10] stiffness of 0.57 N/m so there may be significant variation from chip-to-chip even in a single manufacturer’s batch.

More recently, one of the authors (RG) has been developing the capability at NIST to run the thermal method [11] using laser Doppler velocimetry (LDV). Using this technique, the stiffness of the reference cantilever used in the MRR was measured as 0.734 N/m ± 0.006 N/m which is only 3 % greater than the nominal value originally used for the MRR. The reference cantilever method values provided in this paper assume the original 0.711 N/m values.

5. Added Mass Method

A second calibration method, the added mass method, was utilized at NIST (laboratory “B”) using the same commercial AFM instrument that was used for the reference method. The method was originally developed by Cleveland [12] to calibrate the spring constant of a cantilever using only the resonance frequency measurement of a series of experiments where small, known, masses are added to the end of the cantilever. The exact procedure is described in detail in Appendix C. The technique requires some skill on the part of the operator to be able to apply and remove tungsten or gold microspheres on a cantilever without damaging it but the precision of the resulting data is quite good. The largest uncertainty in the overall process lies with the estimation of the added mass which depends mostly on the measurement of the diameter of the spherical mass added. We typically use a calibrated optical microscope with digital image capture capabilities to estimate both the sphere diameter and the actual location of the sphere on the cantilever (for the offset correction explained in Appendix C). It should be noted that the spring constant estimated with the added mass method is the intrinsic one and no angle correction is needed.

One significant advantage of using the added mass method is that it does not require touching the test cantilever tip to a surface during calibration. Its major drawback is the complexity of the process and the skill required to carefully place microspheres onto the test cantilever surface and remove them without damaging the cantilever.

The results obtained on all six test cantilevers are compared to the reference cantilever method results in Fig. 2. The relative uncertainty of the added mass method was estimated at ± 6 % and is typical for the authors’ experience with this method.

Fig. 2

Comparison of added mass and reference cantilever methods for laboratory “B”.

The results of the two methods agree well statistically and reinforce the previous observation that the six test cantilevers are very similar in spring constant and that the measured calibration values are about 30 % less than the nominal values reported by the manufacturer. Even though the two data sets are statistically equal, there is a consistent trend of the Reference Cantilever data being slightly less in stiffness than the added mass data for all six samples that points to a consistent bias. This can be partially explained by the use of the nominal 0.711 N/m reference cantilever calibration value used for the MRR. If the value of 0.734 N/m (obtained by LDV Thermal) is used instead, the data corrects upward by 3 % and the gap between the two data sets decreases by 50 %. While the scope of the MRR was limited to looking at the precision of the calibration methods and not the accuracy, the numerical agreement of the two methods was a positive sign that these two techniques may also be accurate.

6. Additional Calibration by Thermal and EFB Methods

Three additional techniques were subsequently used to estimate the spring constants of two of the VAMAS test cantilevers. One method, the thermal method, as implemented in an AFM, was performed using two different commercial AFM systems. A Veeco Multimode AFM (Veeco Instruments, Santa Barbara, CA) which was also used for the added mass and reference cantilever methods in the MRR is described as AFM1. A second commercial instrument—an Asylum MFP-3D (Asylum Research, Santa Barbara, CA) standalone is described in this paper as AFM2. The second spring constant calibration technique was an experimental version of the thermal method we have been developing at NIST that utilizes laser Doppler vibrometry—LDV (MSA500, Polytec USA, Hopkinton MA) to measure the power spectrum for the flexural resonance mode of the cantilever. The third calibration technique used was the electrostatic force balance (EFB) [13]. This instrument, designed and developed at NIST, is capable of measuring nanonewton forces applied to surfaces, and can measure spring constants with both good precision and accuracy, since it is SI traceable.

The thermal methods are all based on the original work of Hutter and Bechhoeffer [11] and later refined by several researchers [14, 15]. Based on the equipartition theorem, the thermal method is an energy balance in which the spring constant is obtained through the potential energy term. The technique relies on the measurement of the frequency spectrum obtained while the cantilever is in thermal equilibrium with its environment. Typically, these thermal vibration amplitudes are quite small, and very sensitive, high speed, electronics are required for accurate measurement of both the frequency and vibrational amplitude.

The thermal methods that were implemented on commercial AFM’s used the standard setting recommended by the instrument manufacturers which included a setting of 1.09 for the “chi” [16] correction factor for the Asylum instrument. This correction factor takes into account the effect of the optical lever detection system used in most AFM instruments which actually measure angle changes in the cantilever end and not absolute deflection. The equation used to calculate the spring constant is: where kB is the Boltzman constant, T is the absolute temperature, and the term represents the mean squared displacement of the first bending mode of the cantilever. The first (0.971) term is a mode correction factor that accounts for the first mode displacement contribution of the cantilevers [14] and is small (only 3 %) compared to the almost 20 % for the chi factor correction. Note that the mode correction factor of 0.971 used represents an ideal case for rectangular cantilevers as a simplification. Stark et al. [17] used finite element analysis to estimate the mode correction factor for a particular commercial triangular cantilever (different from the one used in this MRR study) and obtained a value that was slightly lower (0.963).

The chi factor setting used for the Veeco Multimode instrument was not stated but based on an application note [18] from the manufacturer it appears to be the same (1.09) value. AFM thermal calibration also requires a force curve be applied to an infinitely stiff surface to determine the optical lever sensitivity once the laser spot is aligned on the cantilever. This determination suffers from the same issues present in the reference cantilever method and in some cases, friction can cause hysteresis in the approach-retract curves and affect the calibration uncertainty. Once the optical lever sensitivity is obtained, precision of this method is usually quite good (repeatability of a percent or two). It should also be pointed out that because the optical lever sensitivity calibrates the vertical deflection of the tilted cantilever it estimates the vertical component of the spring constant (the “effective” spring constant). This value must be multiplied by cos2 φ (where φ is the inclined angle of the test cantilever) to provide the intrinsic spring constant.

The thermal method being developed on a laser Doppler velocimeter at NIST does not require touching a surface to conduct a force curve since it is already calibrated for deflection in the z direction. It also does not require mounting or tilting of the cantilever so actual cantilever handling can be eliminated and there is no tilt correction component to the calculation. As a result, the mode correction factor [14] is the only adjustment necessary to the spring constant calculated (about 0.97 or 3 % downward adjustment). The repeatability of the measurement varies slightly with type of cantilever but is approximately ± 2 %. The authors are currently cross checking the spring constants obtained using LDV Thermal and EFB to determine the actual accuracy of the LDV thermal method.

The EFB method [13] was developed at NIST to provide SI traceable nanonewton force measurements. The key components are an extremely sensitive electrostatic force transducer combined with an interferometer which allow simultaneous measurement of ultra-small force and displacement as a surface (a flat diamond) is pressed against the measurement feature (in this case the tip of the cantilever). Other key aspects of the instrument include isolation from noise and environmental influences. The instrument is housed in a vacuum chamber, located in a ± 0.1° C temperature controlled NIST metrology laboratory, twelve meters underground. This facility is part of the NIST Advanced Measurement Laboratory—AML. The only major drawback to the EFB method is that it requires touching the AFM tip onto a surface. Since measurements are performed in vacuum, the meniscus forces which can be substantial at these force scales are minimized.

The results of all of the spring constant measurement techniques are summarized for two VAMAS MRR test cantilevers (#3 & #5) in Fig. 3. Since the EFB is SI traceable, it provides a benchmark for the absolute accuracy for each technique for these particular cantilevers. First, it becomes apparent that the actual spring constant of each cantilever is much less than the manufacturer’s nominal value which confirms that nominal values are just estimates and may be off by large amounts. Second, it is encouraging that all of the techniques used provide reasonably similar estimates of the spring constants for each cantilever. If the uncertainties were relaxed to ± two standard deviations (95 % confidence limits) most of the results would be statistically similar. Under tighter scrutiny it appears that the results of the EFB and the LDV thermal are identical and indicate that the LDV method as applied to these cantilevers is both precise and accurate.

Fig. 3

Comparison of spring constants estimated for all methods used at NIST for two test cantilevers (#3 and #5 from the VAMAS MRR).

If we take the EFB value as the benchmark for accuracy we can tabulate the results and provide both a precision (Type A or statistical random uncertainty) and accuracy (as a deviation from the accurate benchmark EFB value and categorized as Type B or systematic uncertainty) estimate for each technique.

Overall, these results demonstrate that for these types of test cantilevers, there are several good methods that can be used to calibrate the spring constant. Each technique has its strengths and weaknesses.

The EFB was used as the accuracy benchmark because it is SI traceable. While it may be the most precise and accurate technique, measurements must be made under vacuum after thermal equilibration which can take days. Data acquisition and analysis is rigorous and time consuming such that it may take a week of experimentation and analysis to produce a single value. The tip actually contacts a surface during the measurement.

For the cantilevers tested, LDV thermal analysis seems to have a precision and accuracy, comparable to that of the EFB. This is especially useful since the LDV thermal method does not require that the tip actually touch a surface during the measurement. Given the much greater ease of use, this technique would seem a very desirable tool in a cantilever spring constant calibration method arsenal. The authors are currently collaborating on investigating the accuracy and precision of this method for a wider range of cantilever types and spring constants to see if the results observed in this study persist.

The AFM instrument Thermal techniques seem reasonably capable of determining spring constants accurately (systematic uncertainty within ± 10 %). The advantage of this technique is that it can be done in the AFM instrument either just before or just after the experimental measurement of interest. We believe that much of the variation in precision and accuracy was due to uncertainties in the optical lever sensitivity measurements, obtained when the cantilever actually touched the surface.

The reference cantilever method was fairly precise (random uncertainty of 5 % to 7 %) and also within 6 % of the accurate value using the manufacturers estimated spring constant of 0.711 N/m. If we instead use the value of 0.732 N/m measured by the LDV Thermal method performed on the reference cantilever the absolute spring constants will adjust upwards 3.2 % resulting in a measurement bias of + 1 % and + 9 % for test cantilevers #3 and #5 respectively.

The Added Mass method was also reasonably accurate (only 4-11 % difference from the benchmark) and precise (relative random uncertainty of 5 %). This method also has the advantage that it does not require the tip to contact the surface during the measurement. This is offset somewhat by the more complex nature of handling microspheres and placing them on the test cantilever.

7. Handling and Use Damage

The initial lesson learned from the MRR was that sharing delicate samples among participants poses dangers to the outcome of the study in several ways. First, these test chips must be physically handled in order to make a measurement in an AFM. This involves picking up the small chips with tweezers and carefully orienting them in the appropriate AFM chip holder and securing them (usually with a small spring clip). There is always the danger that participants can accidently drop the test sample which would usually break the cantilever and then all future data from that sample would be unavailable. There is also danger that the mere act of squeezing the test sample chip with the tweezers can cause fracture damage to the chip and create debris which can settle on part of the cantilever and affect the measurement in a variety of ways, from changing the resonance frequency to changing the reflectivity of the laser on the back of the cantilever. Second, damage can be introduced on use either through wear or accidental contact during alignment or calibration.

In the MRR, the reference cantilever was supplied pre-mounted on a steel puck to eliminate the need to directly handle that particular chip. Despite this precaution, inspection of the reference cantilever at NIST after all of the testing revealed that two of the three original reference cantilevers on the handle chip (ones not actually used for calibration in the MRR) had been broken off during use. One feature of AFM’s is that there is usually a limited view of the intended point of interaction and if there are other cantilevers on the same chip (especially longer cantilevers), they can be inadvertently contacting surfaces out of view. It was thought that additional cantilevers on the chip that were not tested may have inadvertently contacted the unused reference cantilevers, breaking them off.

Inspection of the test cantilevers after the MRR also revealed significant chipping damage on the edge of the chip and significant amounts of debris particles on surfaces of the chip (Fig. 4). Additional scanning electron microscopy imaging of the cantilevers themselves confirmed that the debris did indeed make its way onto the cantilever as well. As the fracture damage seemed to be caused by stress concentrations imposed by the forceps during handling of the chip some suggestions are offered to try to reduce these effects. It is suggested, for example, that future handling procedures specify a particular type of forceps with a flat paddle end that does not produce as high a stress concentration on the sides of the chip during handling as pointed forceps. In addition, a method is suggested for removing the test chips from the adhesive gel used for transportation by gently twisting, rocking, and pealing the chip off the gel. This method should reduce the forces needed to extract the chips from the storage case and will ultimately reduce the amount of debris generated during sample handling. Inspection of the tips of the test cantilevers at NIST after the MRR using field emission scanning electron microscopy showed tip wear and debris attachment to the end of the cantilever. This was likely due to the contact between the tip and surface during force curve measurements necessary for the reference cantilever method. The observation suggests that one potential issue in a wider round robin might be the effect of a changing tip morphology (tip wear) on calibration results. As more participants test the same cantilever, this cumulative damage effect may become more significant. It is also anticipated that sharper Si cantilevers may be more sensitive to this effect, therefore procedural limitations (e.g., limiting the amount of force or the stroke length actually applied during force curve testing) should be implemented to limit cumulative damage from a large round robin among many participants.

Fig. 4

Handling damage on AFM chip used in the mini round robin.

8. Conclusions and Recommendations for a Future Round Robin

There are several results from this study that reveal important information about the potential accuracy and precision of the spring constant measurement techniques used as well as suggestions for improving handling and reporting that could streamline future round robins in this field.

As far as improving the conducting of future round robins, there are several recommendations. They are summarized here in bulleted form as:

Include flat bladed tweezers in test “kit” to reduce chip handling damage

Break off unused cantilevers from test chips

Use chip rocking/twisting method of removal from storage gel

Report to three or more significant figures

Electronic spreadsheet format with consistent calculation documentation should be used to reduce the possibility of transcription and calculation errors.

In addition, systematic characterization of the sample and reference cantilevers prior to and after testing may help document the effects of a large number of participants on the validity of round robin results on such methods where small scale changes may have considerable influence. Optical micrographs at several scales and resonance frequency measurements of the cantilevers are suggested as monitoring tools.

The number of test cantilevers (six) used in the MRR study was, in retrospect, excessive and increased the workload while offering little additional insight. It is suggested that the number of primary samples be reduced to one or two in future studies with the additional focus being put onto providing a wider range of cantilever types (material, shape, size, range of spring constants) that cover the needs of the community. While this MRR was conducted without loss of either test or reference cantilevers (at least the ones that counted), it is anticipated that a wider round robin with more participants would increase the likelihood of accidental damage to the samples and thought should therefore be given to providing “backup” specimens (both test cantilever and reference artifacts) “just in case” such that participants later in the study are given their chance to contribute to their full potential.

All of the spring constant calibration methods performed well for this type of cantilever which was a silicon nitride triangular cantilever with a spring constant of approximately 0.4 N/m. Not only did all the techniques have adequate precision (under ± 7 %), they all agreed within 11 % of an SI traceable benchmark value. The reference cantilever method has the potential for SI traceability and we are currently microfabricating a production batch of reference cantilever arrays [5] that would have their accuracy benchmark established through linkage to EFB measurements.

The thermal method seems very capable of providing precise spring constants with reasonable ease of use. In the case of the cantilevers tested, the AFM methods are accurate to within 10 %. An LDV thermal technique, currently under developmentat NIST, demonstrated a precision and accuracy similar to the SI traceable EFB technique for the cantilever tested. We are currently exploring the validity of these measurements over a wider range of cantilevers using LDV Thermal and EFB to see if it the observed accuracy continues to persist.

A larger round robin currently underway in VAMAS TWA29 will look at expanding the results of this MRR by providing a wider range of cantilevers (different size, shape, material, tip, and spring constant) in order to provide a fuller picture of the capabilities of these different calibration techniques among different laboratories around the world.

Table B1

Raw data reporting format for reference cantilever method

Cantilever ID	ΔLtip μm	φ	Cos2 φ	Ramp size nm	Test #	Srigid V / nm	Scant V / nm
DNP 1A	5.0	11°	0.964	300		Approach or Retract	Approach Retract
					1	0.0234	0.0145
					2	0.0221	0.0151
					3	0.0229	0.0157
					4	0.0235	0.0142
					5	0.0226	0.0146
					6	0.0238	0.0153

Table B2

Calculated data reporting format for reference cantilever method

Test #	L μm	ΔLtip μm	kref N/m	ktest N/m
	429	5.0	0.736
1				0.436
2				0.329
3				0.326
4				0.465
5				0.389
6				0.394
			Avg	0.390
			Std Dev.	0.056
			RSD, %	14.3

Table C1

Raw data reporting format for added mass method

Cantilever ID	L μm	-Sphere -material	Sphere number	Sphere diameter, μm	Resonance Frequency -kHz	ΔLm μm
DNP 6A	108	Gold	–	0	62.8	–
		19300	1	3.8	56.0	+1.9
		kg·m−3	2	5.7	46.8	−1.4
			3	9.5	31.2	−1.1
			–	0	62.8	–

Table 1

Comparison of precision and accuracy for different calibration techniques using Test Cantilever #3

Calibration Technique	Average k (N/m)	Type A Uncertainty†(± 1 sd, %)	Comparison to Benchmark‡(%)
Added Mass	0.392	5.4	+4 %
Reference Cantilever	0.370	6.8	−2 %
LDV Thermal	0.3781	1.8	0 %
AFM Thermal (A)	0.410	4.4	+9 %
AFM Thermal (B)	0.381	3.9	+1 %
EFB	0.3764	1.6	0 %(benchmark)

Also termed random uncertainty (obtained using statistical methods)

Relative to EFB benchmark

Table 2

Comparison of precision and accuracy for different calibration techniques using Test Cantilever #5

Calibration Technique	Average k(N/m)	Type A Uncertainty†(± 1 sd, %)	Comparison to Benchmark‡(%)
Added Mass	0.427	5.2	+11 %
Reference Cantilever	0.407	5.2	+6 %
LDV Thermal	0.3808	1.6	−1 %
AFM Thermal (A)	0.365	1.6	−5 %
AFM Thermal (B)	0.360	1.0	−6 %
EFB	0.3841	1.4	0 % (benchmark)

Also termed random uncertainty (obtained using statistical methods)

Relative to EFB benchmark