Nonlinear Dynamics of Cantilever Tip-Sample Surface Interactions in Atomic Force Microscopy (2010)

The various dynamical implementations of the atomic force microscope have become important nanoscale characterization tools for the development of novel materials and devices. One of the most significant factors affecting all dynamical AFM modalities is the cantilever tip-sample surface interaction force. We have developed a detailed mathematical model of this interaction that includes a quantitative consideration of the nonlinearity of the interaction force as a function of the cantilever tip-sample surface separation distance. The model makes full use of cantilever beam dynamics and the multiply differentiability of the continuous force-separation curve that results in a set of coupled differential equations, Eqs.(14) and (15), for the displacement amplitudes of both the cantilever and the sample surface. The coupled dynamical equations are recast in matrix form and solved by a standard iteration procedure, but space limitations allow only a presentation of the salient features of the procedure. Although the mathematical form of the coupled equations are valid for any vibrational mode, only flexural vibrations of the cantilever and out-of-plane oscillations of the sample surface are considered. We emphasize that Eqs.(14) and (15) are obtained assuming that the cantilever is a rectangular beam of constant cross-section, the dynamics of which are characterized by a set of eigenfunctions that form an orthogonal basis for the solution set. For some other cantilever shape a different orthogonal basis set of eigenfunctions would be appropriate. However, the mathematical procedure used here would lead again to Eqs.(14) and (15) with values of the coefficients appropriate to the different cantilever geometry. Practicably, this means that the shape of the cantilever is not as important in the solution set as knowing the cantilever modal resonant frequencies, obtained experimentally. The modal frequencies and solution set are expanded to include nonlinear modes generated by nonlinear interaction forces or large cantilever drive amplitudes. A general steady state solution of the coupled dynamical equations is found that accounts for the positions of the excitation force (e.g., a piezo-transducer) and the cantilever tip along the length of the cantilever and for the position of the laser probe on the cantilever surface. The solution is applied to two dynamical AFM modalities - resonant difference-frequency atomic force ultrasonic microscopy, and the commonly used amplitude modulation-atomic force microscopy. Image generation and contrast equations are obtained for each of the two A-AFM modalities assuming for expediency that the contrast results only from variations in the sample stiffness constant. Since the sample stiffness constant is related directly to the Young modulus of the sample, the contrast can be expressed in terms of the variation in the Young modulus from point to point as the sample is scanned. We note further the existence of two values of the sample stiffness constant, corresponding to the dominantly attractive and dominantly repulsive regimes of the force-separation curve. The two values allow for a bi-stability in the cantilever oscillations that is experimentally observed. Equations for both the maximum nonlinearity regime and the hard contact (linear) regime of cantilever engagement with the sample surface are obtained. For dynamical AFM operation outside these regimes, it is necessary to use all terms in the solution set given in Section 2 to describe the signal output of a given A-AFM modality. The extent to which the hard contact (linear regime) equations apply depends on how well the approximation F(z 0 ) - holds.