Calibration of the Spring Constant

To obtain quantitative force versus distance information, precise knowledge of the cantilever spring constant is a prerequisite. In principle, values of the cantilever spring constants are given by the manufacturers, but these values are often not very 

Spring constants of AFM cantilevers can be calculated from the dimensions and material properties of the cantilever according to Eq. (3.4). For triangular cantilevers, as a first approximation, it can be viewed as the combination of two parallel rectangular beams. If w is the width of one arm, the spring constant is 205. 

More complex analytical equations have been deduced by several authors [207, 213-215] that take into account deviations both from the ideal V-shape and from the thin plate theory. Numerical calculations using finite element allow an even more realistic simulation of shapes and bending behavior [207, 214, 216, 217]. For a recent discussion of these different dimensional approaches and their refinement, see Refs [218, 219]. 

Calculation of spring constant from cantilever dimensions and material properties suffers from two main problems - exact determination of cantilever thickness is difficult and Young s modulus of the thin layer of cantilever material may differ from the bulk value - and thus, calculated spring constants were found to significantly deviate from actually measured ones [207, 215]. The possible existence of reflective coatings on the backside of the cantilever further complicates the calculations. Therefore, experimental determination for each single cantilever is usually employed. 

One method is the thermal noise method introduced by Hutter and Bechhoefer [220]. It is implemented in many commercial AFMs. The thermal noise method is based on the equipartition theorem of statistical mechanics that states that the mean thermal energy of any harmonic system at temperature T is equal to ke T/2 per degree of freedom. For the small thermal oscillation amplitudes ( 0.1 nm), the AFM cantilever can be seen as a harmonic oscillator with spring constant k. The mean square deflection of the cantilever due to thermal fluctuations must fulfill the following condition  

---

Quantitative evaluation of a force-distance curve in the non-contact range represents a serious experimental problem, since most of the SFM systems give deflection of the cantilever versus the displacement of the sample, while the experimentalists wants to obtain the surface stress (force per unit contact area) versus tip-sample separation. A few prerequisites have to be met in order to convert deflection into stress and displacement into tip-sample separation. First, the point of primary tip-sample contact has to be determined to derive the separation from the measured deflection of the cantilever tip and the displacement of the cantilever base [382]. Second, the deflection can be converted into the force under assumption that the cantilever is a harmonic oscillator with a certain spring constant. Several methods have been developed for calibration of the spring constant [383,384]. Third, the shape of the probe apex as well as its chemical structure has to be characterised. Spherical colloidal particles of known radius (ca. 10 pm) and composition can be used as force probes because they provide more reliable and reproducible data compared to poorly defined SFM tips [385]. 

---