As far as I am aware, AFM systems must be calibrated for each new probe to obtain an accurate conversion between the deflection of the cantilever dz and the commensurate force applied to the cantilever F. A systematic explanation of the conversion steps that are needed, references to various methods to calibrate the spring constant of the probe, and analytical expressions showing how uncertainties of the calibration of the dz propagate to uncertainties on F are given in my paper here.

Article An uncertainty budget analysis on the Hamaker constant deter...

The angular deflection of a calibrated probe causes a linear offset of a laser beam on a sensor. Conversion from angular deflection to linear offset is done using the small angle approximation whereby tan(theta) ~ theta. To be truly accurate, either the sensor should be a curved surface or the deflection should be converted using the proper tan(theta) in post-processing by software.

As you note, when you move the laser spot on the probe, you also change the factor needed to make the conversion from linear offset on a sensor to angular deflection of the probe (to force applied to the probe).

Hi Jeffrey! Cheers for the answer and I'll give your paper a good read over the next couple of days. So the small angle approximation is applied to the optical lever to deduce the rotational deflection alongside calibration to determine the individual probes spring constant. However, I'm still confused how the linear offset on the photodetector and rotational angle find the vertical deflection of the probe.

To what I understand the following is the method used. For the optical lever, the following equation is formed based upon trigonometry:

α=∆d/2L

Where α, ∆d and L are the angular deflection, change in distance on the photodetector and length between the photodetector and probe respectively. This is then utilised to calculate the rotational deflection. From this, the Euler-Bernoulli theory is applied to convert to vertical deflection of the probe. To my understanding it follows the subsequent equation:

∆z= 2αl/3

Where ∆z and l are the change in vertical deflection and length of the probe. If this is correct, therefore the length of the probe or the point along its length the laser reflects off is required.

I have only seen the above spelled out so clearly and to greater depth in "Piezoresistive AFM cantilevers

surpassing standard optical beam

deflection in low noise topography

imaging" by M.Dukic et al. However, how the length is known or determined by an AFM (alongside the length between the photodetector and probe for that matter) is not detailed in the above or anything to what I can find.

FIrst, with what you have, recast the two equations as below with \Delta d as the movement on the detector, \Delta z as the deflection of the probe of length L_p, and L_D as the distance between the probe and the detector.

\Delta d = 3 \Delta z (L_D / L_p)

The magnification factor (L_D / L_p) is what generates the sub-nanometer sensitivity in height.

Now consider how we self-calibrate an AFM to set \Delta d to reflect a change in height of the sample. Hold the probe at a distance Z above the surface and allow the laser to hit exactly at the center of the detector. Scan laterally over a feature with a height increase of dZ. The laser moves to a position D. Move the probe or the surface using a calibrated piezo by a distance -dZ to return the laser back to the position at the center. Clearly by this, you have established that \Delta d (a shift in laser spot position) = dZ (a change in height on the sample).

The calibration to obtain F from \Delta d is outlined in my paper. I believe the original work from T J Senden is the true starting point (and I think that method was the basis perhaps of some parts of the Digital Instruments patents at the time). Otherwise, I have never seen any further thorough treatment of the factors that are involved in doing the F

Ah I see! So this is essentially when in the contact regime from the force-distance curve and finding the voltage/deflection-to-distance gradient for the sample (ideally a hard sample relative to the probe stiffness)?

Ah intriguing! It's interesting in your paper to see such a thorough analysis of the factors and so uncertainties involved in the deflection and force determination with some I have never considered alongside your method. Think it would be useful to apply such an uncertainty methodology to some of my work!

The motivation for my questions is related to exploring and quantifying the thermo-mechanical behaviour seen in bi-material probes (like most AFM probes) with a specific emphasis on SThM probes. I have theoretical and FEA models of these probes that I have seen a strong correlation in the temperature-to-deflection relationship to my experimental work when a probe is out-of-contact. As a result, I am trying to see if I can similarly see this from the laser signal when out-of-contact and then a difference for in-contact. Therefore, I need to fully understand how the voltage change on the photodetector relates to the probes deflection and work it back (as how thermo-mechanical behaviour manifests along a probe is different to an end force) to see if it is reading what I am expect for the temperature changes seen.

As you continue, feel free to contact me off-line for further comments.