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Q on spring constants of stiff cantilevers

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dalia posted on Tue, Dec 21 2010 8:56 PM


I heard that in general thermal tune method to calibrate the spring constant of stiff cantilevers is not very reliable (this has an impact on QNM calibration.)  Why is this the case?  If you get good signal to noise in the thermal tune spectrum and a good fit, shouldn't the spring constant that you get from this method be OK?  I understand that stiff cantilevers don't oscillate as strongly as soft cantilevers.  What stiffness spring constant are thermal tunes considered reliable for? Thanks, Dalia

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Suggested by Ben Ohler


As you correctly state, the accuracy of the thermal-tune method decreases as cantilevers become stiffer. This happens because the thermal motion decreases (kz^2/2=kbT/2) with increasing k, to the point where the motion falls below the sensitivity of the detector.

This obviously happens first in the regions with low response (off resonance) but quite often you can still fit a curve to the peak even though the background is dominated by the sensor noise. A spring constant value can be derived from this fit, but it will be less accurate.

To answer your question, if you get a good tune you should be able to get a good fit, and a spring constant to the accuracy of the measurement. Ben Ohler, Bruker’ internal expert on this subject, has written a good Application Note and Paper on this subject. Linked here:


Ohler (Rev Sci Instr, 2007).pdf

The cantilever stiffness ranges that you can use thermal tune for are subjective and depend on your calibration requirements, but as a rule of thumb I would recommend only for k less than 10N/m with the AFM. If you switch to a vibrometer then k <100N/m.


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Suggested by Ben Ohler

Hi Dalia,

I'll add a bit to what Steve has already written.

My RSI paper that he linked is a good starting point. All of the AFM data in it was taken on a MultiMode. As you note, it really comes down to signal to noise ratio. As that term isn't well defined in this context, I presented my own definition in the paper. I looked at the ratio of the area under the resonance peak (the "signal") divided by the area under the baseline. I made this measurement over a relatively small bandwidth, about 15X the width of the resonance at half max. This is intended to approximate how you actually make the thermal tune calibration, which does occur within a smaller region of the spectrum near the resonance. Comparing Figures 1 and 2, you can see that the thermal tune results remained consistent with the other measurements until the S/N dropped to <1 for the ~29 N/m cantilever measured on the AFM (S/N=0.128). This cantilever calibrated fine on the vibrometer (S/N=2.41) and the softer ~3N/M probe calibrated fine on the AFM (S/N=3.22). So while I can't offer an exact answer, I would suggest that the S/N (when measured in this same way) should remain >2 or 3 for the most reliable results.

As Steve notes, unless the noise gets extremely high, the resonance peak will still be visible above the baseline noise. But the curve fit will not converge stably unless you give it some baseline data to fit. So I'm not convinced that you can effectively recover the peak shape obscured by the baseline noise. I really think you need to depend on baseline noise being small in comparison to the measured thermal noise.

You should be able to make this S/N measurement on all of our systems that have thermal tune as well as on competitor's systems. On our systems all of the curve fit parameters are given in the dialog box, from which you can calculate the peak area and the area under the baseline. You shouldn't base your decision about whether the measured spring constant is reliable by judging whether or not it is reasonable versus the nominal value. The sort of ~20% error shown in Figure 1C is well within the normal variance of spring constants from nominal values. Likewise, it's not very useful to compare with a different AFM since it won't be obvious which one is correct. For the greatest confidence in the calibration you really need to scrutinize the measurement carefully and ideally compare it to other techniques.

It helps to keep in mind that the thermal tune method was first and most widely used for soft cantilevers, <1 N/m. The majority of the literature out there referencing the technique is using these softer probes. So the accuracy of the method for stiffer probes has not been widely investigated. Our development of the PeakForce QNM technology drove us to stiffer probes and higher scrutiny of the spring constant calibration. For softer probes the thermal tune method is great and quite reliable, but for stiffer probes it is important to be aware of the fundamental limitations of the method and the AFM hardware.



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