Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer

Abstract

The nonlinear equations of motion for a silicon cantilever beam, covered by a piezoelectric lead–zirconate–titanate layer, subjected to a Lennard-Jones type boundary condition, are derived for voltage excitation. The Lagrangian of the system is obtained from the electric enthalpy density, including the virtual work of the Lennard-Jones potential, assuming the beam undergoes only small displacements. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to different tip to sample distances. An analytically obtained frequency response function enables determination of the frequency shift of individually piezoactuated microbeams, which are proposed as fundamental elements of parallel atomic force microscopy, undergoing forced vibration in a dissipative environment.

Nonlinear mathematical modeling of vibrating motion of nanomechanical cantilever active probe

“…The required geometric information and mechanical properties are provided in Table 1. The coefficients of Lennard-Jones are selected as σ=0.34(nm) and H=10 -18 (J) (Wolf and Gottlieb, 2002). Natural frequency of discontinuous NMC can be calculated through boundary conditions equations, continuity of deformation, slope, bending moment, shearing force, and taking the coefficients determinant equal to zero.…”

“…The theoretical calculations which were made with regard to the discontinuous beam method for NMC show that the first natural frequency is equal to 50.6(kHz) with only 1.2% error in proportion to the practical results. Wolf and Gottlieb (2002) examined actuator piezoelectric cantilever with throughout piezo layer and uniform cantilever near the sample surface. Taking into account the piezoelectric actuator moment and the force between the tip and the sample at the boundary conditions of motion differential equation, they offered another formulation for frequency response.…”

“…The total potential energy of NMC can now be formulated as follows: in which, superscript (t) indicatestip. The force imposed by the surface of the sample to the tip can be expressed according to Lennard-Jones model (Wolf and Gottlieb, 2002) as follows:…”

“…Here, unlike earlier works in which NMC has been studied in contrast with sample surface (Wolf and Gottlieb, 2002;Fung and Huang, 2001)and with throughout piezoelectric layer and horizontal to the sample surface, piezoelectric layer has been piecewise, and it is supposed to be angled to the sample surface, and the effects of piezoelectric confining electrodes in modeling would not be ignored. To do so, NMC is modeled with the help of continuous beam theory of Euler Bernoulli and considering the existing discontinuities in the length to increase the accuracy.…”

“…Most of the mathematical models which have been used until the date are lumped-mass spring models (Sarid et al, 1997;Pishkenari et al, 2008), while it has been proved (Nayfeh et al, 1992) that nonlinear lumped mass models which approximate continuous dynamic systems with the nonlinear boundary conditions may encounter substantial errors. NMC vibrating analysis has been studied at the specific attachment of piezoelectric (throughout layer) with simple rectangular cantilever and considering the continuous beam model in non-contact mode (Wolf and Gottlieb, 2002;Fung and Huang, 2001). …”