A finite element formulation for nanoscale resonant mass sensing using the surface Cauchy–Born model

Abstract

The purpose of this work is to develop the theoretical basis needed to study nanoscale resonant mass sensing with finite elements using the surface Cauchy–Born (SCB) model. The theory is developed in 1D, where it is identified that the primary modeling issue lies in capturing inhomogeneous surface stresses arising from adsorbate/substrate interactions. By utilizing internal degrees of freedom within the SCB framework, we show that the SCB model can represent the bonding energies, and thus the inhomogeneous surface stress that arises due to interactions by atoms of dissimilar materials. A key outcome of this is that it is shown that a finite element solution using the SCB model is able to simultaneously capture both mass and stiffness variations due to adsorbate/substrate interactions, and their effects on the nanostructure resonant properties. We first verify that the SCB model accurately captures the resonant properties of monatomic 1D atomic chains, then demonstrate the approach by studying the resonant properties of 1D atomic chains that interact with adsorbates. Importantly, we demonstrate that a finite element solution using the SCB model can predict the distinct shifts in resonant frequency that occur due to the adsorption of different materials on the 1D monatomic chain.