Casimir Arc Plate Geometry: Computational Analysis of Thickness Constraints for Gold and Silver Nanomembranes in MEMS Applications

TL;DR

This work analyzes curvature reversal of a finited, curved nanomembrane (arc) facing a plate under Casimir forces within a MEMS context. It develops a computational framework combining the Proximity Force Approximation with next-to-leading-order corrections and Kirchhoff-Love bending energy to determine a critical membrane thickness via $U_{Casimir} > U_{Bending}$. Results indicate reversal occurs at nanoscale thicknesses, with silver membranes tolerating slightly greater thickness than gold, and demonstrate that NTLO-corrected PFA provides physically reasonable bounds over the distance range $d \in [0.1,1]\,\mu$m. These findings yield conservative design constraints for MEMS and point to experimental validation and 3D extensions as fruitful future directions for Casimir-based actuation and stiction prevention.

Abstract

A theoretical analysis of the Casimir interaction between an arc and plate is conducted, which remains unexplored despite its relevance to Micro-Electro-Mechanical Systems (MEMS) fabrication. The configuration consists of a rigid finite plate and a flexible curved nanomembrane, with radius 100 micrometers, initially concave toward the rigid plate. The maximum thickness is evaluated for which the nanomembrane undergoes a change in curvature: from concave to convex with respect to the plate, due to the Casimir interaction. The Casimir energy for a curved surface is derived using the Proximity Force Approximation (PFA) with next-to-leading-order (NTLO) corrections. Kirchhoff-Love theory for a thin isotropic plate of constant thickness is used to estimate the bending energy. Material-dependent effects on the Casimir interaction are evaluated by comparing Au and Ag plates. The maximum thickness is derived where U_Casimir > U_bending for distances in the range of 0.1-1 micrometers. Results show curvature reversal occurs for nanomembranes with nanoscale thicknesses at the studied distances. Silver nanomembranes tolerate greater thickness than gold nanomembranes due to material-dependent properties. Comparison between NTLO-corrected PFA and perturbative PFA confirms the accuracy of the NTLO approach. The Casimir arc-to-plate geometry in MEMS enables Casimir-based actuation, enhances devices reliability, and prevents stiction. These findings provide thickness constraints for MEMS design and performance, accounting for the Casimir force.

Figures (4)

• Figure 1: Arc-to-plate geometry at distance (a) $0.1 \mu m$ (b) $1 \mu m$.
• Figure 2: Casimir energy vs distance for various radii in a sphere-to-plate geometry.
• Figure 3: Maximum thickness of Au and Ag nanomembranes: corrected PFA with NTLO for the distance range of (a) $0.1 \le d \le 1 \mu m$.(b) $0.1 \le d \le 0.2 \mu m$.
• Figure 4: Fractional deviation between the leading-order PFA and the PFA with NTLO correction as a function of separation distance for (a) for Au (b) for Ag.