Lateral, normal, and longitudinal spring constants of atomic force microscopy cantilevers

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Abstract
For a V-shaped atomic force microscopy cantilever beam, the spring constants in the three principal directions are given in terms of the beam geometry and material properties. For the lateral stiffness, a closed-formed expression is presented. Also, the normal and the longitudinal stiffness are obtained from a few simple equations. The results are compared with a finite element study and found to be very accurate. All spring constants depend strongly on the cantilever thickness, which is difficult to measure. In addition, the lateral and longitudinal stiffness are sensitive to the location and the height of the attached pyramid.