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Diamagnetic Lateral Force Calibrator (D-LFC)

Lateral force calibration of AFM (LFM, FFM)

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Basic of AFM system

Deformation characteristics of AFM cantilever

Principles of D-LFC

Basics of AFM system [top]

Scanning probe microscopes (SPMs) are a family of instruments used for studying surface properties of materials from the atomic to the micron level. Usually they contain the components illustrated in Figure 1. Depending on the different tip-sample interaction and detection mechanisms, there are dozens of variations of SPMs. Among them, the Atomic Force Microscopes (AFM) is perhaps the most popular application due to its versatility of handling nano-scale business. 

SPM schematic

Figure 1. Schematic of SPMs (adapted from http://www.park.com)

The most commonly used AFM system consists of a cantilever-tip assembly and its force transducers. The tip is typically made of a silicon or silicon nitride apex formed by chemical etching, or a bead of a few micrometers in radius, sometimes called a colloid tip, attached to a thin silicon or silicon nitride film cantilever. The transducer is commonly made of a position sensitive photo detector (PSPD) array, as shown in Figure 2, which senses the deflection of a laser beam reflected off the top surface of the AFM cantilever near the end. The deflection of the laser beam is generated by the bending and torsional deformation of the cantilever, which is caused by normal contact and friction forces exerted by the substrate material surface against the tip. 


Figure 2. Force sensing of an AFM system


Deformation characteristics of AFM cantilever [top]

To have a quantitative measurement, we need to relate the PSPD output signals: vn and vl (in the unit of volts),  to their physical origins: normal force n and lateral force f (in the unit of newtons). Intuitively, for a linear system, we need four quantities to convert vn, vl to nf.

eq1aeq1b            (1)

eq2aeq2b          (2)

For a long time, people had idealized the situation by ignoring the two terms  aln and anl. Apparently this is not correct and significant deviations were observed and reported by some researchers, which they called "cross-talk" effect. In order to get a better assessment of the forces involved in AFM system, many methods were proposed to minimize or compensate the crosstalk. However, as far as we know, most of approaches didn't provide a clear and generic analysis on the deformation characteristics of AFM system; therefore the calibration methods they proposed are quite limited and sometimes even wrong. 

Generally speaking, the crosstalk can be induced mechanically, i.e. the shear center misalignment of the cantilever with respect to the geometric center of the cantilever, and optically, i.e. the misalignment of the PSPD array. For a properly aligned AFM system, the optical misalignment is minimized and the crosstalk coefficient anl is negligible compared to aln

Our analysis shows that: 

  • the force constants all and ann are determined by the cantilever spring constants (normal and lateral) and the PSPD sensitivity factors
  • the crosstalk force constant aln depends on the mechanical characteristics of the cantilever-tip assembly (such as the spring constants of the cantilever, shear center of the cantilever along the contacting plane, the tip location and shape, etc), as well as the optical alignment and detection

Detailed mathematical descriptions could be found in our recent paper on the journal of Review of Scientific Instruments.

Principle of D-LFC [top]

Now the main goal is how we figure out a reliable way to calibrate the three force constants  aln, all and ann.  According to the above discussion, all the approaches could be cataloged into two types: direct method or indirect method.

Calibrating the constants based on the measurement of the individual intrinsic properties of the AFM system, is an indirect method. Evidently, the indirect method has three error sources. One is in the estimation of the torsional spring constant of the cantilever, another in assessing the effective cantilever-tip structural geometry parameters, and the last in gauging the angle sensitivity constant of the PSPD. In many cases, the indirect method erroneously determines the force constants to be an order of magnitude away from their actual values. Thus, calibration of the force constants based on an indirect method is very difficult and ineffective.

Calibrating  aln, all and ann based on (1) and (2), regarding them as constants of system response, is a direct method of calibration. For this approach, a reference spring with a known spring constant is needed. Our invention of the D-LFC makes it possible for the first time.

As discussed above, the principle of the D-LFC is very simple: a pyrolytic graphite sheet levitated by a strong magnetic field is used as a reference spring to apply a known force on the AFM cantilever-tip assembly and by recording the output signals the force constants can be obtained as a system response.



DIY Steps

Lateral Calibration