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

Mario Versaci,Luisa Fattorusso,Paolo Di Barba

doi:10.3390/math10010054

Mario Versaci, Luisa Fattorusso + Show 1 more

Open Access

https://doi.org/10.3390/math10010054

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Journal: Mathematics	Publication Date: Dec 24, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: University of Reggio Calabria, University of Pavia

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.

Highlights

MEMS with Fringing Field: AIn recent years, scientific research has paid particular attention to the study of model descriptions with different levels of accuracy and detail, and to the behavior of microelectro-mechanical systems (MEMSs) [1,2]
Many models have been formulated for electrostatic MEMSs with parallel metallic plates considering all the physical-technical specifications [1,2,13,14]
Unlike the works known in the literature concerning the physical-mathematical modeling of MEMS with parallel plates [1], it is necessary to take into account the phenomenon of the fringing field, which is quite frequent in some industrial MEMS devices and whose main problem is the excessive deformation of the deformable element, with the consequent risk of touching the upper plate that would cause unwanted electrostatic discharges; should these

Summary

Introduction

Scientific research has paid particular attention to the study of model descriptions with different levels of accuracy and detail, and to the behavior of microelectro-mechanical systems (MEMSs) [1,2] They are devices of various kinds (mechanical, electrical, and electronic) integrated in a highly miniaturized form onto the same substrate of semiconductor material (for example, silicon [3,4,5]) that combine the electrical properties of the semiconductor with the opto-mechanical properties. The following Section offers a brief overview of the genesis of the mathematical model of the parallel plate MEMS device, with particular reference to devices affected by the fringing field.

Genesis of the Model with the Fringing Field

Advantages and Disadvantages of the Model Under Study

Statement of the Global Existence Theorem

Some Physical Considerations on the Statement of Theorem 1

Proof of Theorem 1

Possible Intended Uses of the MEMS Device Under Study

Conclusions

Some Future Perspectives