Force gradient detection

Fig. 15.5. Force-gradient detection via mechanical resonance. Experimental and calculated resonance frequencies of a cantilever for various positions of the tunneling tip. Upper family of curves, for the copper, and lower, for the steel beam. (Reproduced from Diirig et al., 1986, with permission.)...

Use Equation VIII-1 to determine the effective mass of the cantilever if the cantilever has a spring constant C = 20 N/m, the minimum detectable force gradient is hF/dz = 4 X 10 N/m, and the frequency shift is 200 kHz. How does the frequency shift depend on distance from the surface if the force has a 1/z distance dependence ... 

Figure 15.6 is a schematic diagram of an AFM with an optical interferometer (Erlandsson et al., 1988). The lever is driven by a lever oscillator through a piezoelectric transducer. The detected force gradient F is compared with a reference value, to drive the z piezo through a controller. In addition to the vibrating lever method, the direct detection of repulsive atomic force through the deflection of the lever is also demonstrated. 

The data in Fig. 11 show an offset between the force and stiffness minima in the approach and retraction curves. The explanation for this is shown in Fig. 12, which illustrates the relationship between potential, force and interaction stiffness. These curves provide a basis for determining where the contact point with the surface is located. One definition of contact is the position on the curve where the repulsive force can first be detected (see 24), typically identified by a change in curvature of the force-displacement data (3). Therefore, the force gradient (stiffness-displacement data) reveals more clearly the attractive to repulsive transition. The initiation of repulsive contact is thus found from the minimum of the stiffness approach curve (marked at 0 nm), which marks the maximum attractive interaction stiffness. The stiffness data represent a convolution of force gradient and contact stiffness and is... 

A variation on the amplitude modulation technique was also used to measure oscillatory surface forces with increased sensitivity in a branched hydrocarbon, squalene. In this technique, the sample was oscillated with low amplitude (c. 1 A), and both the cantilever static and dynamic (induced oscillation from a change in the tip-sample force gradient) deflection was measured. Figure 1.13 shows the static force measurement and Fig. 1.14 the dynamic measurement, shown as an interaction stiffness. The sensitivity of the dynamic force measurement is such that the interdigitation of the branched methyl groups can be detected (indicated by arrows in Fig. 1.14). 

The second application of noncontact AFM is in near-field detection of force gradients at 5-50 nm above the surface. This is the approach used in electric force microscopy (EFM) [158] and magnetic force microscopy (MFM) [159]. 

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