Determine Cantilever Spring Constant by Thermal Tune

1. Determine cantilever deflection sensitivity if you have not done so already (see Contact Mode Force Calibration).

A correction factor should be applied to this calibration. When we determine the deflection sensitivity, we exert a static load at the end of the cantilever, but when we measure the thermal energy the cantilever is oscillating at resonance and is constrained at only one end rather than two. The cantilever bends differently in the two situations and thus reflects the laser beam of the optical lever differently. See D. Sarid, “Scanning Force Microscopy With Applications to Electric, Magnetic and Atomic Forces,” 2nd Edition, Oxford University Press, New York 1994, pp 5–6 and pp 10–11. When you calibrate the deflection sensitivity using a static force, you underestimate the amplitude of dynamic motion by approximately 8% (for a simple beam). This correction factor is applied using the Deflection Sensitivity Correction parameter in the Thermal Tune window.

2. Click Withdraw.

3. Select Calibrate > Thermal Tune or the Thermal Tune icon in the NanoScope tool bar

4. Select a frequency range over which you will tuneThe 5–2000 kHz range is not available on the NanoScope V-PI.1.

5. Click Acquire Data in the Thermal Tune window:

Figure 1: The Thermal Tune window

6. The microscope will acquire data for about 30 seconds.

7. A power spectral density (PSD) plot of the cantilever response to ambient conditions is displayed. Activate either the Lorentzian (Air) or Simple Harmonic Oscillator (Fluid) radio button to select a the appropriate model to fit to the data.

Click to view the equations used to fit the filtered data Simple Harmonic Oscillator, for use in fluid: where: A(ν ) = amplitude as a function of frequency, ν A0 = baseline amplitude ADC = amplitude at DC (zero frequency) ν 0 = center frequency of the resonant peak Q = quality factor Lorentzian, for use in air: where: A(ν ) is the amplitude as a function of frequency, ν A0 is the baseline amplitude ν 0 is the center frequency of the resonant peak C1 is a Lorentzian fit parameter C2 is a Lorentzian fit parameter

8. Adjust the Median Filter Width parameter in the Thermal Tune window to remove individual (narrow) spikes. This replaces a data point with the median of the surrounding n (n =3, 5, 7) data points.
9. Adjust the PSD Bin Width parameter to reduce noise by increasing the averaging.
10. Drag markers in from the left and/or right plot edges to bracket the bandwidth over which the fit is to be performed.

Typically the markers are located roughly where the spectrum rises from the noise floor. Precise placement is unnecessary; the fit is insensitive to the minimal power contributed from these frequencies far from the natural resonance (see right “shoulder” of waveform in Figure 1). You may exclude a larger portion of a shoulder of the waveform from the fit bandwidth to ignore a noise spike. Experiment with repeated fits of the same acquired thermal tune data to become familiar with its sensitivity to bandwidth and choice of model.

11. Click the Fit Data button. The curve fit, in red, is displayed along with the acquired data. If necessary, adjust the marker positions and fit the data again to obtain the best fit at the thermal peak.
12. Adjust the markers for the bandwidth of the fit PSD to include in the spring constant calculation.

13. Enter the cantilever Temperature (in °C).
14. Click the Calculate Spring K button. You will be asked whether you want to accept the calculated value of the spring constant, k:

Clicking Yes copies the calculated spring constant in the “Ramp” list in the PicoForce view.