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Analysis of a Model for Electrical Discharge in MEMS

Heiko Gimperlein, Runan He, Andrew A. Lacey

Abstract

We study the local well-posedness of the solution to a coupled nonlinear elliptic-parabolic system which models electrical discharge in a Micro-Electro-Mechanical System (MEMS). A simple MEMS capacitor device contains two plates acting as the capacitor's electrodes, one of which is flexible, and which are separated by a narrow gas-filled gap. In the event of the flexible plate approaching the other, electrical discharge can occur. This is modelled here by two parabolic equations, for densities of electrons and positive ions, and an elliptic equation for electric potential. We show the local-in-time existence of a weak solution for the coupled system. Compactness techniques, used previously in the study of drift-diffusion equations, are employed in our proof.

Analysis of a Model for Electrical Discharge in MEMS

Abstract

We study the local well-posedness of the solution to a coupled nonlinear elliptic-parabolic system which models electrical discharge in a Micro-Electro-Mechanical System (MEMS). A simple MEMS capacitor device contains two plates acting as the capacitor's electrodes, one of which is flexible, and which are separated by a narrow gas-filled gap. In the event of the flexible plate approaching the other, electrical discharge can occur. This is modelled here by two parabolic equations, for densities of electrons and positive ions, and an elliptic equation for electric potential. We show the local-in-time existence of a weak solution for the coupled system. Compactness techniques, used previously in the study of drift-diffusion equations, are employed in our proof.
Paper Structure (13 sections, 10 theorems, 176 equations, 1 figure)

This paper contains 13 sections, 10 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Physical Motivation
  3. Literature Review
  4. Overview and Structure of the Paper
  5. Preliminaries
  6. An Auxiliary System and A Priori Estimates
  7. A Priori Estimates for the Auxiliary System
  8. Existence of a Solution to the Auxiliary System
  9. Positivity of Solutions to the Auxiliary Problem
  10. Proof of Theorem \ref{['mainthm']}
  11. Proof of Theorem \ref{['mainthm']}: Existence
  12. Proof of Theorem \ref{['mainthm']}: Uniqueness
  13. Proof of Theorem \ref{['Exist_Aux_Pro']}

Key Result

Theorem 1.1

Under Assumption main_assum below, there exists a unique weak solution of the problem EPSys-bdyval on a time interval $(0, T_1)$ and which satisfies $p\geq 0$ and $n\geq0$ almost everywhere.

Figures (1)

  • Figure 1: Sketch of a MEMS capacitor undergoing sparking. The shaded region is $\Omega$; the electrodes $A$ and $B$ constitute the Dirichlet part of the boundary, $\partial\Omega_D$, on which electric potential $\phi$ is known and the concentrations $p$ and $n$ are assumed known; the artifical boundary $C$ is supposed sufficiently distant for homogeneous Neumann condtions to apply, so $C = \partial\Omega_N$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.6
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4: Aubin-Lions Lemma
  • Remark 2.5
  • ...and 11 more