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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.

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The goal in this effort is twofold: ͑1͒ to develop an understanding of Casimir forces in geometries more complicated than the usual parallel-plate geometry and ͑2͒ to provide extensive numerical computations to elucidate quantitative and qualitative aspects of the vacuum fluctuation energy and Casimir forces for the rectangular cavity. We review geometries for which Casimir forces and vacuum energy have been computed, and point out some of the difficulties with the ideal-conductor boundary conditions and ideal-shape boundary conditions, e.g., infinitely sharp edges. We investigate the vacuum electromagnetic stress-energy tensor at 0 K for a perfectly conducting three-dimensional rectangular cavity with sides a 1 ϫa 2 ϫa 3 . The elements of the tensor are averaged over the appropriate spatial coordinates of the cavity. We first consider the average energy density T 00 ϭe(a)/V from the viewpoint of symmetry, where e(a 1 ,a 2 ,a 3 )ϭe(a) is the finite change in the zero-point energy from the free-field case. The vacuum energy e(a) and the total vacuum force on the wall normal to the i direction, F i ϭϪ‫ץ‬e/‫ץ‬a i , are both homogeneous functions of the cavity dimensions. Because of this symmetry, the energy and forces are related by the equation e(a)ϭa•F(a). We compute the vacuum forces and energy numerically for cavities with a broad range of dimensions. The implications of the perfectconductor boundary conditions and the effects of the edges of the cavity are both considered. The C 3v symmetry of the constant-energy surfaces is apparent. The zero-energy surface, which is invariant under dilations and therefore extends to infinity, separates the nested, concave, positive-energy surfaces from the open, negative-energy surfaces. The positive-͑negative-͒ energy surfaces are mapped into each other by scale changes. The force F(a) is normal to the constant-energy surface at a. The surfaces corresponding to zero forces, F i (a)ϭ0, are invariant under dilations and are therefore infinite. The zero-energy surface and the zero-force surfaces delineate the different geometries for which there are zero, one, or two negative ͑inward or attractive͒ forces on the cavity walls, along with the sign of the corresponding energy. There is no rectangular cavity geometry for which all forces are negative or zero; conversely, only geometries that are not too different from a cube have all positive ͑outward or repulsive͒ forces. Only for the last case is the energy e(a) necessarily positive. To provide an intuitive feeling for these vacuum energies, comparisons are made to other forms of energy in small cavities. We consider the energy balance for changes in cavity dimensions.

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