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TECHNICAL PAPERS

# Analysis of One-Dimensional and Two-Dimensional Thin Film “Pull-in” Phenomena Under the Influence of an Electrostatic Potential

[+] Author and Article Information
Gang Duan

Mechanical Engineering,  University of Missouri-Rolla, Rolla, MO 65409-0050

Kai-Tak Wan

Mechanical Engineering,  University of Missouri-Rolla, Rolla, MO 65409-0050wankt@umr.edu

Note that Ref. 19 normalizes $w$ and $V0$ using $g$ instead of $h$ as in the present work. It will become apparent that normalization using $h$ will lead to a better description of the mixed bending-stretching bridge behavior.

J. Appl. Mech 74(5), 927-934 (Sep 18, 2006) (8 pages) doi:10.1115/1.2722311 History: Received April 27, 2006; Revised September 18, 2006

## Abstract

A thin one-dimensional rectangular or two-dimensional axisymmetric film is clamped at the perimeter. An electrostatic potential $(V0*)$ applied to a pad directly underneath the film leads to a “pull-in” phenomenon. The electromagnetic energy stored in the capacitive film-pad dielectric gap is decoupled from the mechanical deformation of the film using the Dugdale-Barenblatt-Maugis cohesive zone approximation. The ratio of film-pad gap ($g$) to film thickness ($h$), or, $γ=g∕h$, is found to play a crucial role in the electromechanical behavior of the film. Solution spanning a wide range of $γ$ is found such that $V0*∝γ3∕2$ for $γ<0.5$ and $V0*∝γ5∕2$ for $γ>5$. The new model leads to new design criteria for MEMS-RF-switches.

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## Figures

Figure 1

Sketch of a typical MEMS-RF switch. The suspended bridge deforms in the presence of an electrostatic force induced by the electrode-pad directly underneath.

Figure 2

Normalized bridge deformed profile as a function of membrane stress. The bridge anchors at ξ=0 and has its midspan at ξ=1. The dashed curves show the plate-bending and membrane-stretching limits.

Figure 3

Mechanical response of the bridge under a uniform pressure across the span. The dashed curves show the plate-bending and membrane-stretching limits.

Figure 4

The gradient n(ω0) of the mechanical response ρ(ω0)

Figure 5

Forces acting on the bridge in the stretching limit, with the attractive electrostatic force shown as dashed curves for a range of applied voltage, and the cubic mechanical force on the bridge shown as dark curve (OACB). Stable equilibrium is maintained along the path OAC. Pull-off occurs at C.

Figure 6

Energetics of the MEMS-RF switch with υ0=1.00 in the stretching limit, showing various energy terms as functions of bridge central displacement

Figure 7

(a) Total energy as a function of central bridge displacement for a range of applied voltage in the stretching limit. Stable equilibrium is maintained along the path OAA′C. Path CB′B is unstable and physically inaccessible. Pull-in occurs at C. (b) Total energy as a function of both central bridge displacement and applied voltage.

Figure 8

Pull-in (w0*∕g) as a function of the bridge-pad gap. Both force and energy balances are shown. Pull-in occurs within the shaded area.

Figure 9

Pull-in voltage as a function of the bridge-pad gap. Both force and energy balances are shown. Pull-in occurs within the shaded area. The dashed lines show the plate-bending and membrane-stretching limits.

Figure 10

Sketch of a 2D axisymmetric MEMS-RF switch

Figure 13

Pull-in voltage as a function of the bridge-pad gap. Pull-in occurs within the shaded area.

Figure 12

Pull-in (w0*∕g) as a function of the bridge-pad gap

Figure 11

(a) Total energy ΣT(ω0,υ0) for fixed υ0 in the stretching limit. Pull-in occurs at C. (b) Total energy ΣT(ω0,υ0).

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