Carbon nanotubes observed in experiments [

18] always contain more than millions of atoms. Consequently, MD has difficulties in studying the feasibility of the proposed NEMS design in practical applications. In this paper, we employ a continuum approach, the mesh-free particle method [

19], to model the memory cell via discretizing the shells of nanotubes as particles. During the simulation, the outer tube is fixed and has no deformation. We first assume that the inner tube is deformable. Therefore, the following equations of motion are solved at each particle on the inner tube:

where *m*
_{I} is the mass associated with particle I, **u**
_{I} is the displacement of particle I, and
is the internal nodal force applied on particle I due to the deformation of the nanotube itself. The external nodal force,
, contains two parts. One is due to the interlayer interaction between the inner tube and the outer tube, and the other is the induced electromagnetic force when applying voltage on the electrodes.

Generally, the Lennard-Jones 6–12 potential [

12] has been employed to describe the van der Waals interaction between shells in a multi-walled carbon nanotube (MWNT) in a molecular model. The potential function is written as

where *A* = 2.43 × 10^{−24} J nm and *y*
_{0} = 0.3834 nm. The interlayer equilibrium distance is 0.34 nm, which results in the minimum van der Waals energy. This distance matches the thickness of a graphene sheet, and it also satisfies the criterion proposed by Legoas et al. [11] for stable nanotube-based oscillators.

In the mesh-free particle model, the major issue is how to calculate interaction between particles at different layers in an MWNT to approximate molecular-level interlayer interaction. To solve this issue, we choose two representative cells of area

*S*
_{0}, each containing

*n* nuclei (

*n* =

*2* in this paper for graphene sheets). The continuum-level van der Waals energy density is defined as

where

is the distance between the centers of those two considered cells. One is on the outer tube, and the other is on the inner tube. Then, the total continuum-level non-bonded energy is calculated as

where Ω_{O}and Ω_{I}are the configurations of the outer and inner tubes, respectively. Then, the force applied on particle I can be derived as the first derivative of Φ with respect to the coordinates of particle I.

It should be noted that Φ is the interlayer potential when atoms are placed at the equilibrium positions. Therefore, interlayer friction due to atoms’ thermal vibration cannot be directly calculated from the continuum approximation. We employ MD to simulate nanotube-based oscillators at the room temperature of 300 K. The interlayer friction, which causes the energy dissipation, is calculated as 0.025 pN per atom. In all, the external force applied due to the interlayer interaction is

where *v*
_{Iz
} is the *z* component of the velocity of particle I, and *N* is the number of atoms represented by particle I in the mesh-free particle model. Here,**e**
_{
z
} represents direction along the nanotube axis.

In the proposed NEMS design, an electrode of potential

*V* with the ground plane that has the zero potential can be viewed as a capacitor. Its capacitance is expressed as

where

E is the electric field and ε

_{0} = 8.854 × 10

^{−12} F/m is the permittivity of free space (in farads per meter). Since the energy stored in a capacitor is

the induced electrostatic force can be calculated as

where

*z*
_{I} is the axial position of the atom on the inner tube. The electromagnetic forces are in the direction of the higher electric field density and therefore serve to localize the inner nanotube underneath the electrode with the higher applied WRITE voltage. We only consider the axial electrostatic forces because: (1) the motion of the inner tube is along the axial direction, and (2) the transverse electromagnetic forces are small enough to be ignored. The classical conductor model is used here to approximate the electrostatic field induced in the proposed NEMS design. Consequently, equations of motion, i.e., Eq.

1, can be rewritten as