A Detailed Analysis of the Resonant Frequency and Sensitivity of Flexural Modes of Atomic Force Microscope Cantilevers with a Sidewall Probe Based on a Non local Elasticity Theory

In this paper, utilizing a nonlocal elasticity theory, the resonant frequency and sensitivities of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) are studied. This ACP comprises a horizontal cantilever and a vertical extension, and a tip located at the free end of the extension, which makes the AFM capable of topography at the sidewalls of microstructures. First, the governing differential equations of motion and boundary conditions for flexural vibration are obtained via a combination of the basic equations of nonlocal elasticity theory and Hamilton's principle. Afterwards, a closed-form expression for the sensitivity of vibration modes is obtained using the relationship between the resonant frequency and contact stiffness between tip and sample. This analysis provide a better representation of the vibration behaviour of an AFM cantilever with a sidewall probe, for which the effects of the small scale are significant. The results of the nonlocal theory are compared to those of classic beam theory. The evaluation shows that the resonant frequency and sensitivity of the proposed ACP are size dependent, especially when the contact stiffness are high.