Nonlinear Multimode Dynamics of a Moving Microbeam For Noncontacting Atomic Force Microscopy

Atomic force microscopy (AFM) is a modern imaging technique that is used to map surfaces down to atomic resolution and enables a quantitative estimation of atomic interaction forces. This is obtained by measuring a van der Waals like atomic interaction between a sample and a vibrating microcantilever, which has a sharp tip at its free end. Of particular importance are biological and nonconducting materials that cannot be mapped by alternative methods, without destruction of their surfaces by a conducting coating layer. The growing demand for detection of sub-atomic features and industrial use, increases the need for faster and more accurate scanning. Various methodologies have been proposed to speed up the scan rate. These include individual moving microbeam control strategies and use of an array of probes. However, the accuracy of force estimation from measured data crucially depends on the quality of the mathematical model in use. A typically used model is that of a lumped mass system that reduces the microbeam to a linear spring with a nonlinear force, derived from a candidate tip-sample interaction. This model does not incorporate the dynamic boundary condition of the scan process and cannot resolve the rich spatio-temporal dynamic response of the nonlinear dynamical system. Thus, the objectives of this research include theoretical derivation and analyses of a continuous model for the moving and vibrating AFM microbeam that consistently incorporates the nonlinear atomic interaction and the dynamic conditions of the scan process. A nonlinear initial boundary-value problem is derived using the extended Hamilton’s principle. The continuum system is then reduced to a multimode dynamical system using a Galerkin procedure. Numerical analysis of a three mode system reveal that below the ‘jump-to-contact’ stability threshold, there exist a dense set of coexisting bounded periodic (ultrasubharmonic) solutions. This complex bifurcation structure is augmented by quasiperiodic solutions that are found to correspond to a 3:1 internal resonance between the third and second microbeam modes.

Jump to contact stability threshold

A quasiperiodic time-series (top) and power spectra (bottom).