Analysis of Stiction in Nanoelectromechanical Systems Using Molecular Dynamics Simulations and Continuum Theory

Abstract

Recognizing that either the retarded van der Walls (sometimes called the Casimir) or the full van der Walls force always acts between two flat surfaces that are in close proximity to each other, we analyze the stiction phenomenon in nanoelectromechanical systems (NEMS) composed of a single layer of graphene suspended over a stationary rigid substrate of graphene by using both molecular dynamics (MD) simulations and a linear elasticity theory. Stiction occurs when the initial gap between the deformable and the stationary electrodes of a NEMS is small enough for the two electrodes to touch each other in the absence of any applied potential difference between them. The value of the initial gap at stiction is called the critical gap and determines the fabricability of the device. In this work, an NEMS is modelled as a pre-stressed clamped atomic graphene structure in the form of either a rectangular strip or a solid/annular circular initially flat disk suspended over a rigid flat substrate and the critical gap found as a function of the prestress. Both methods involve different challenges – the MD work requires properly estimating the Casimir force and the continuum problem has a deflection-dependent external force. It is shown that results from the two approaches qualitatively agree with each other. For the continuum problem we provide a simple expression for estimating limiting dimensions of the NEMS.

Appendix

The virial stress on an atom $i$ is given by [21]:

$$ \boldsymbol{\sigma _{i}} = \frac{1}{\varPi _{\mathrm{i}}} \biggl(- m_{i} \dot{\boldsymbol{u}_{i}} \otimes \dot{\boldsymbol{u}_{i}} + \frac{1}{2} \sum _{j\neq i} \boldsymbol{x}_{ij} \otimes \boldsymbol{f}_{ij} \biggr). $$

(A.1)

In Eq. (A.1) $m_{i}$ is the mass of atom $i$, $\boldsymbol{x_{i}}$ its present position, $\dot{\boldsymbol{u_{i}}}$ its present velocity, $\boldsymbol{x}_{ij} = \boldsymbol{x}_{i} - \boldsymbol{x}_{j}$, ⊗ denotes the tensor or the dyadic product between two vectors, $\varPi _{\mathrm{i}}$ is the volume assigned to atom $i$, and $f_{ij}$ equals the interatomic force between atoms $i$ and $j$. We calculate the prestress as the average of the virial stresses for all atoms in layer $A$, e.g., see Cormier et al. [22].

$$ \boldsymbol{\sigma} _{0} = \frac{\sum _{i} \varPi _{i} \boldsymbol{\sigma }_{i}}{\mathcal{A} h}. $$

(A.2)

The software LAMMPS outputs the value $\varPi _{i} \sigma _{i}$ for each atom $i$. To avoid assigning a value to $h$ for a single-layer graphene sheet [23], we plot the initial critical gap against the quantity ‘$\sigma _{0} h$’ for both the MD simulations and the analysis of the linearly elastic problem.