The role of the casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS)

Abstract

We present an analysis describing how the Casimir effect can deflect a thin microfabricated rectangular membrane strip and possibly collapse it into a flat, parallel, fixed surface nearby. In the presence of the attractive parallel-plate Casimir force between the fixed surface and the membrane strip, the otherwise flat strip deflects into a curved shape, for which the derivation of an exact expression of the Casimir force is nontrivial and has not been carried out to date. We propose and adopt a local value approach for ascertaining the strength of the Casimir force between a flat surface and a slightly curved rectangular surface, such as the strip considered here. Justifications for this approach are discussed with reference to publications by other authors. The strength of the Casimir force, strongly dependent on the separation between the surfaces, increases with the deflection of the membrane, and can bring about the collapse of the strip into the fixed surface (stiction). Widely used in microelectromechanical systems both for its relative ease of fabrication and usefulness, the strip is a structure often plagued by stiction during or after the microfabrication process—especially surface micromachining. Our analysis makes no assumptions about the final or the intermediate shapes of the deflecting strip. Thus, in contrast to the usual methods of treating this type of problem, it disposes of the need for an ansatz or a series expansion of the solution to the differential equations. All but the very last step in the derivation of the main result are analytical, revealing some of the underlying physics. A dimensionless constant, Kc, is extracted which relates the deflection at the center of the strip to physical and geometrical parameters of the system. These parameters can be controlled in microfabrication. They are the separation w0 between the fixed surface and the strip in the absence of deflection, the thickness h, length L, and Young’s modulus of elasticity (of the strip), and a measure of the dielectric permittivities of the strip, the fixed surface, and the filler fluid between them. It is shown that for some systems (Kc>0.245), with the Casimir force being the only operative external force on the strip, a collapsed strip is inevitable. Numerical estimates can be made to determine if a given strip will collapse into a nearby surface due to the Casimir force alone, thus revealing the absolute minimum requirements on the geometrical dimensions for a stable (stiction-free) system. For those systems which do exhibit a stiction-free stable equilibrium state, the deflection at the middle of the strip is always found to be smaller than 0.48w0. This analysis is expected to be most accurately descriptive for strips with large aspect ratio (L/h) and small modulus of elasticity which also happen to be those most susceptible to stiction. Guidelines and examples are given to help estimate which structures meet these criteria for some technologically important materials, including metal and polymer thin films.