Electrostatic-Elastic MEMS with Fringing Field: A Problem of Global Existence

Abstract

In this paper, we prove the existence and uniqueness of solutions for a nonlocal, fourth-order integro-differential equation that models electrostatic MEMS with parallel metallic plates by exploiting a well-known implicit function theorem on the topological space framework. As the diameter of the domain is fairly small (similar to the length of the device wafer, which is comparable to the distance between the plates), the fringing field phenomenon can arise. Therefore, based on the Pelesko–Driscoll theory, a term for the fringing field has been considered. The nonlocal model obtained admits solutions, making these devices attractive for industrial applications whose intended uses require reduced external voltages.

Finite Differences for Recovering the Plate Profile in Electrostatic MEMS with Fringing Field

“…Whatever the intended use of the device, it is necessary that there are no electrostatic discharge phenomena inside it, caused by contact between deformable and fixed elements, which would damage the device itself [14][15][16]. Therefore, it appears necessary to reduce, as much as possible, the physical causes, such as the fringing field, to produce an excessive approach between deformable and fixed elements [17][18][19]. The fringing field, which strongly depends on the length/width ratio of the device, produces important effects on the bending of the lines of force of the electric field, E, inside it, manifesting this influence near the edges; however, in the center, this effect is almost nil [20,21].…”

“…However, due to its formulation, the dimensionless model studied in [22] is not very suitable for technology transfer, since some of its analytical elements do not correctly model the behavior of some elements present in the device. This impasse was overcome in [17] where, starting from [22], a new formulation of the dimensionless model was presented and studied, more in keeping with the industrial reality of the MEMS devices produced, which also considers the effects due at the fringing field. This dimensionless model takes the form:…”

“…Ω represents the smooth bounded domain (MEMS device); • u : Ω → R defines the profile of the deforming plate; • f : Ω → R + is a bounded function which considers the dielectric properties of the material constituting the deformable plate; • λ is a positive dimensionless parameter depending on the applied voltage, V, defined as in (3) which represents the ratio of a reference electrostatic force to a reference elastic force; • β is a positive dimensionless parameter which takes into account the stiffness of the deformable plate; • γ is a positive dimensionless parameter which takes into account the stretching effect in the deformable plate; • χ, defined as in (8), is a positive dimensionless parameter which takes into account the non-local dependence of V on the solution due to a possible non-uniform electric charge distribution; • δ is a positive dimensionless parameter (usually constant) that weighs the effects due to the fringing field. • λδ|∇u(x)| 2 takes into account, according to the Pelesko-Driscoll theory, the effects due to the fringing field [17,23].…”

“…Model (1) appears interesting for many reasons. On one hand, λδ|∇u(x)| 2 allows easy software/hardware implementations, while on the other hand, it is voltage controllable (presence of λ that, as specified in (3), depends on V), even if the dependence of δ on the applied external voltage has not yet been highlighted (satisfactory limitations of δ have been obtained for membrane MEMS devices [17,23,24]). However, even if the stability of (1) has not yet been studied (for which any unstable positions of the deformable plate would risk causing unwanted electrostatic discharges), the device is controlled in voltage (presence of λ).…”

“…We also observe that, in (1), there are terms due to stiffness and self-stretching (identifiable by the parameters β and γ) of the deformable element to take into account the mechanical phenomena of fatigue due to the continuous rising-lowering of the deformable element. Furthermore, with the deformation of the plate, significant variations of electrostatic capacitance occur inside the MEMS, making it necessary to express its dependence on the geometric parameters of the device and on an auxiliary electrostatic capacitance (C f ) capable of opposing sudden voltage variations applied (presence of χ in (1)) [1,17]. Finally, f (x) in (1) (present in the "capacitive" term of the model) models the dielectric properties of the plates.…”