A 2D Membrane MEMS Device Model with Fringing Field: Curvature-Dependent Electrostatic Field and Optimal Control

Abstract

An important problem in membrane micro-electric-mechanical-system (MEMS) modeling is the fringing-field phenomenon, of which the main effect consists of force-line deformation of electrostatic field E near the edges of the plates, producing the anomalous deformation of the membrane when external voltage V is applied. In the framework of a 2D circular membrane MEMS, representing the fringing-field effect depending on ?|∇u|2 with the u profile of the membrane, and since strong E produces strong deformation of the membrane, we consider |E| proportional to the mean curvature of the membrane, obtaining a new nonlinear second-order differential model without explicit singularities. In this paper, the main purpose was the analytical study of this model, obtaining an algebraic condition ensuring the existence of at least one solution for it that depends on both the electromechanical properties of the material constituting the membrane and the positive parameter δ that weighs the terms ?|∇u|2. However, even if the the study of the model did not ensure the uniqueness of the solution, it made it possible to achieve the goal of finding a stable equilibrium position. Moreover, a range of admissible values of V were obtained in order, on the one hand, to win the mechanical inertia of the membrane and, on the other hand, to ensure that the membrane did not touch the upper disk of the device. Lastly, some optimal control conditions based on the variation of potential energy are presented and discussed.

A Semi-Linear Elliptic Model for a Circular Membrane MEMS Device Considering the Effect of the Fringing Field

“…To overcome its mechanical inertia, the applied voltage V must assume values such that the corresponding value of the electrostatic field inside the device can generate an appropriate electrostatic pressure equal to (with , the permittivity of free space). The term can be translated into an equivalent electrostatic force, , computable as [ 1 , 25 , 48 , 61 ] which deflects the membrane, thus achieving a displacement in its center, , equal to (where T is the radial mechanical tension of the membrane when it is at rest) [ 1 , 48 ]. Electrostatically, if the membrane deforms, the field , which depends on the distance between the membrane and the upper disk, results to be locally orthogonal to the tangent line to the membrane at the same point [ 2 ].…”

“…Moreover, many researchers are actively engaged in the development of important experimental research works for the development and prototyping of special MEMS such as, for example, circular graphene membrane MEMS devices [ 43 , 44 ], SiN circular membrane MEMS devices [ 45 , 46 ], and CMOS MEMS-based membrane-bridge devices [ 47 ] particularly useful for industrial applications. Moreover, the scientific community is intensively working on the analysis/synthesis of multi-physical models characterized by a high degree of symmetry, because these are more easily achievable from a technological point of view [ 24 , 30 , 48 , 49 , 50 , 51 ]. Accordingly, we have focused our attention on a 2 D circular membrane MEMS device, a kind of geometry widely used in many industrial applications [ 24 , 25 , 48 , 52 ].…”

“…Moreover, the scientific community is intensively working on the analysis/synthesis of multi-physical models characterized by a high degree of symmetry, because these are more easily achievable from a technological point of view [ 24 , 30 , 48 , 49 , 50 , 51 ]. Accordingly, we have focused our attention on a 2 D circular membrane MEMS device, a kind of geometry widely used in many industrial applications [ 24 , 25 , 48 , 52 ]. The membrane deformation is described by its profile, , with r being the radial coordinate and R the radius of the device.…”

“…As proved in [ 48 ], the field is locally orthogonal to the straight-line tangent to the membrane profile itself, so that it is natural to consider with mean curvature of the membrane [ 24 ]: Considering both ( 6 ) and ( 7 ), ( 4 ) can be rewritten as which is a 2 D second-order differential semi-linear elliptic model without explicit nonlinearity. The term , the so-called critical security distance, guarantees that the membrane does not touch the upper disk.…”

“…In contrast, in [ 25 ], the equilibrium configurations were analyzed. Finally, in [ 48 ], the model was improved and rewritten. In particular, in this study, the effect of the fringing field on membrane curvature has been considered.…”