On the Partial Differential Equations of Electrostatic MEMS Devices: Stationary Case

Abstract

We analyze the nonlinear elliptic problem ∆u = λf (x) (1+u) 2 on a bounded domain Ω of R N with Dirichlet boundary conditions. This equation models a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 above a rigid ground plate located at −1. When a voltage -represented here by λ-is applied, the membrane deflects towards the ground plate and a snap-through may occur when it exceeds a certain critical value λ * (pull-in voltage). This creates a so-called "pull-in instability" which greatly affects the design of many devices. The mathematical model lends to a nonlinear parabolic problem for the dynamic deflection of the elastic membrane which will be considered in forthcoming papers [11] and [12]. For now, we focus on the stationary equation where the challenge is to estimate λ * in terms of material properties of the membrane, which can be fabricated with a spatially varying dielectric permittivity profile f . Applying analytical and numerical techniques, the existence of λ * is established together with rigorous bounds. We show the existence of at least one steady-state when λ < λ * (and when λ = λ * in dimension N < 8) while none is possible for λ > λ * . More refined properties of steady states -such as regularity, stability, uniqueness, multiplicity, energy estimates and comparison results-are shown to depend on the dimension of the ambient space and on the permittivity profile.

Touchdown and related problems in electrostatic MEMS device equation

“…First, the above profile of C r is observed numerically [8]. Next, some estimates on λ are given for nonradially symmetric case [8,9]. Finally, existence of the non-minimal non-radially symmetric stationary solution is proven by the variational method [4].…”

“…First, the above profile of C r is observed numerically [8]. Next, some estimates on λ are given for nonradially symmetric case [8,9].…”

“…The structure of the set of stationary solutions has been studied for p = 2 ( [8,11]). We obtain, similarly, an upper bound of λ for the existence of the solution denoted by λ > 0 for general p > 1.…”

On the partial differential equations of electrostatic MEMS devices II: Dynamic case

“…By using both analytical and numerical techniques, they obtained larger pull-in voltage λ * and larger pull-in distance for different classes of varying permittivity profiles. These results were extended and sharpened in [9], where we focussed on the steady-state solutions of (1.1), i.e,…”

On MEMS equation with fringing field

“…• There exists a critical number λ * > 0 such that for 0 < λ < λ * problem (S λ ) has a minimal stable solution u λ , while for λ > λ * there are no solutions to (S λ ) (see [6]). …”