Comment on “Print your atomic force microscope” [Rev. Sci. Instrum. 78, 075105 (2007)]

In the Figure 3 of their paper Kühner et al.¹ obtained that the MLCT-D cantilever (Bruker, USA) in water and out of contact vibrates with the mean deflection amplitude of about 3 nm at a frequency of its first flexular resonance (∼1.4 kHz in their case), and of about 0.6 nm in its static limit (taken at a frequency 100 Hz). As we show below their deflection noise scale in Fig. 3 cannot be correct. We also speculate that similar mistakes apply to the Figure 5 (in their paper), which presents the cantilever's thermal noise spectrum in contact with a substrate. An amplitude of the cantilever's vibrations in contact with any substrate is tricky to compare, since it depends on many factors, such as the contact force, type of the substrate, its surface roughness, etc.^2–4 Thus, we cannot comment on their Figure 5, but we are in a position to comment on their Figure 3.

In Figure 1, we produce vibration spectra for thermally excited MLCT-C and MLCT-D levers obtained using our custom made AFM. First, we sampled a raw (250 kHz analog low pass filtered at 48 dB/octave) cantilever deflection (A-B) signal from a photodiode. A total of 248 data sets with N = 2¹⁵ points each at a sampling frequency of 625 kHz. The A-B data sets are Fourier transformed⁵ into the power spectra with a frequency spacing (also known as frequency resolution), Δν, of 625 kHz/(2¹⁵) = 19.1 Hz.⁶ The power spectra are averaged. The averaged power spectrum is converted into the root mean square (RMS) power spectrum by dividing it by N² and by a factor of 2 (see Ref. 5). The RMS power spectrum is converted into the power spectral density, PSD, by dividing it by Δν. The PSD is converted into the units of nm²/Hz using voltage sensitivity S (in [V/nm]). The value of S (also known as an inverse of a “slope”) is obtained from initial portion of the cantilever's contact with a stiff sample.⁷ It represents an amount of bending of a cantilever recorded by a photodiode (in [V]) due to a corresponding scanner displacement (in [nm]). Two corrections are applied to S. First, the value of S is obtained in the scanner's reference frame, but the forces act in the cantilever's reference frame, which is tilted at a repose angle φ with respect to the sample.⁸ Thus, we need to multiply the value of S by the cos(φ).^8–10 Our φ = 18 ± 1° (as found from the interference pattern).¹¹ Second, the value of S is acquired with the cantilever end-loaded, while the PSD is obtained for a freely oscillating lever. The cantilever has slightly different shapes in these two situations.^8,12 In the limit of a small laser spot positioned on the end of a cantilever, the value of PSD needs to be multiplied by a correction factor γ = 1.124.^13,14 We obtain the displacement noise density, x_RMS, in the units of nm/Hz^1/2 as a square root of the PSD. To compare with the results of Kühner et al., we plot (in Fig. 1(c)) the x_RMS in (nm), which is obtained from x_RMS in (nm/Hz^1/2) by multiplying the latter by a square root of Δν. Equation (1) summarizes our results:

Main results of Figure 1 are as follows. First, we note that the MLCT-D levers are typically about twice as stiff as the MLCT-C levers. Thus, we expect the static mean deflection of the MLCT-D levers to be roughly (neglecting higher order effects) two times less than in the case of the MLCT-C levers. This is what we get by comparing Figs. 1(a) and 1(b). Second, comparing various batches of the MLCT cantilevers we found very similar displacement noise density spectra for the respective types of levers (within 20% tolerance, data not showed here). Third, our results match the predictions of the equipartition theory. The equipartition theory postulates that for each resonance mode of an AFM cantilever the amount of fluctuations squared, 〈δx_RMS²〉, multiplied by a respective elastic spring constant for a given mode, k, must equal to the thermal energy k_BT, where k_B is the Bolzmann's constant, and T is the absolute temperature.¹⁵ In summary: k 〈δx_RMS²〉 = k_BT. The value of 〈δx_RMS²〉 is obtained from an integrated PSD [nm²/Hz] around each particular resonance, such that: 〈δx_RMS²〉 = Integrated (PSD [nm²/Hz]). In the case of a first flexular resonance for used here MLCT-C lever, we get 〈δx_RMS²〉 = (0.27 ± 0.02) nm². Similarly, for a MLCT-D lever, we obtain: 〈δx_RMS²〉 = (0.098 ± 0.004) nm². Typical values of k for the first resonance modes are ∼ 15 pN/nm for MLCT-C levers and ∼ 40 pN/nm for MLCT-D levers. These yield k 〈δx_RMS²〉 = 15 pN/nm × 0.27 nm² ∼ 4.0 pN nm and k 〈δx_RMS²〉 = 40 pN/nm × 0.098 nm² ∼ 3.9 pN nm, respectively. Such results agree with the value of k_BT ∼ 4.1 pN nm at room temperature. Finally, our results agree with the results presented in the literature for these types of cantilevers, e.g., results in Table II in the Ref. 9, and in Figure 1 in the Ref. 16.

Kühner et al. probably used an older incarnation of the MLCT-D cantilevers than we did. However, based on our experience and on what others published in the last years,^11,16 we speculate that the MLCT-D levers of Kühner et al. had very similar mechanical properties to the currently available ones. Second, the cantilever's deflection measurements are subjected to some variability due to mechanical and geometrical differences between different batches of the AFM cantilevers. There are also calibration issues related to various AFM designs, laser light positioning on the cantilever, size of the laser beam, and other issues described in detail in the Refs. 12 and 15. Most of these issues, however, cancel out when a voltage sensitivity, S, is obtained as we did.¹⁷ Overall, similar calibrated vibrational spectra are obtained in different labs for the same types of levers when using the slope calibration method. Why then did Kühner et al. get about 60–100 times higher deflection noise amplitudes? We speculate that several numerical errors have compounded in the paper of Kühner et al. First, instead of multiplying their initially measured deflection noise density, in nm/Hz^1/2, by a square root of their frequency resolution Δν (around (2 Hz) ^1/2 as judged roughly from the first five points on the frequency scale in Fig. 3 of their paper), they multiplied their deflection noise by a square root of their bandwidth. Second, they happened to choose the bandwidth incorrectly, as determined by a first resonance of their cantilever ∼1.4 kHz (as seen in their Fig. 3). Third, as a relatively minor issue, they plausible did not use any correction factors for the value of their voltage sensitivity S. The data of Kühner et al. would match our results in Figure 1 when recalculated using Eq. (1).

As related, we discuss the issue of bandwidth. The cantilever is a complicated beam with many resonance modes.¹⁸ Figure 1 shows a cantilever's response with its many non-attenuated resonance modes limited only by electronics. Such “electronics bandwidth” (230 kHz in Fig. 1) comes chiefly from the circuitry in the photodetector.¹⁹ Why then is the cantilever's resonance frequency (∼1 kHz in Figure 1) confused with the AFM bandwidth?

Electronics bandwidth turns out to be of a limited use in the context of the speed of the cantilever's reply. From Fig. 1, one can see that the maximum amplitude of the cantilever's mechanical response to external vibrations is achieved at a frequency of its first flexular resonance mode. Using current fast speed piezoelectric positioner, it is the cantilever's first resonance, which determines speed of the cantilever's response. This situation applies in the force extension (FX) AFM mode,^15,20–22 where an AFM cantilever is approached/retracted from the substrate by a piezoelectric positioner. In another AFM force spectroscopy mode, the force clamp (FC) AFM mode,²³ the cantilever is held under a constant deflection via an external feedback working on the piezoelectric positioner. Then, it is not the cantilever's resonance frequency (∼1 kHz) anymore, but an inverse of its damping coefficient (expressed in 1/s), which determines the time necessary for the cantilever to respond to any changes. This is because any mechanical oscillator responds the fastest in its critically damped conditions. Such conditions, however, arise only with a great care in the feedback adjustments.

Overall, the rectifying effects on force and displacement measurements in the FX, FC AFM modes arise from the cantilever and the AFM feedback (if applies) and set the “mechanical AFM bandwidth.” Consequently, one should distinguish between “electronics” and “mechanical” bandwidths. The electronics bandwidth relates to the amplitudes of the cantilever's thermal noise, while the mechanical bandwidth defines the speed of the cantilever's response.

The work was supported by the National Science Foundation (NSF) under Award No. EPS-0903806 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.