Curvature-Dependent Electrostatic Field as a Principle for Modelling Membrane-Based MEMS Devices. A Review

Abstract

The evolution of engineering applications is increasingly shifting towards the embedded nature, resulting in low-cost solutions, micro/nano dimensional and actuators being exploited as fundamental components to connect the physical nature of information with the abstract one, which is represented in the logical form in a machine. In this context, the scientific community has gained interest in modeling membrane Micro-Electro-Mechanical-Systems (MEMS), leading to a wide diffusion on an industrial level owing to their ease of modeling and realization. Physically, once the external voltage is applied, an electrostatic field, orthogonal to the tangent line of the membrane, is established inside the device, producing an electrostatic pressure that acts on the membrane, deforming it. Evidently, the greater the amplitude of the electrostatic field is, the greater the curvature of the membrane. Thus, it seems natural to consider the amplitude of the electrostatic field proportional to the curvature of the membrane. Starting with this principle, the authors are actively involved in developing a second-order semi-linear elliptic model in 1D and 2D geometries, obtaining important results regarding the existence, uniqueness and stability of solutions as well as evaluating the particular operating conditions of use of membrane MEMS devices. In this context, the idea of providing a survey matures to discussing the similarities and differences between the analytical and numerical results in detail, thereby supporting the choice of certain membrane MEMS devices according to the industrial application. Finally, some original results about the stability of the membrane in 2D geometry are presented and discussed.

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

“…Note that . This result has been proved in [ 23 ] and confirmed numerically in [ 30 ]. On the other hand, due to reasons of symmetry, the points of characterized by the greatest slope of the membrane profile can only be those located on .…”

“…We can also observe that the greater the , the greater the curvature of the membrane. On this basis, we have that can be considered locally proportional to the mean curvature of the membrane [ 23 , 24 , 25 , 30 ].…”

“…However, such theoretical models hardly provide analytical solutions. Hence, it appears necessary to have conditions that can ensure both the existence and uniqueness of the solution [ 23 , 24 , 25 , 26 , 30 ] to avoid ghost solutions when the model is solved numerically [ 30 , 31 , 32 ]. Since the aforementioned analytical solutions are unobtainable, suitable numerical procedures have to be selected, and it becomes imperative to evidence the pros and cons of each method [ 25 , 26 , 30 , 31 , 32 ].…”

“…Hence, it appears necessary to have conditions that can ensure both the existence and uniqueness of the solution [ 23 , 24 , 25 , 26 , 30 ] to avoid ghost solutions when the model is solved numerically [ 30 , 31 , 32 ]. Since the aforementioned analytical solutions are unobtainable, suitable numerical procedures have to be selected, and it becomes imperative to evidence the pros and cons of each method [ 25 , 26 , 30 , 31 , 32 ]. MEMS is a rampant technology employed for realizing thermo-elastic systems [ 1 , 33 , 34 , 35 , 36 , 37 ], and it has a wide variety of applications ranging from biomedical engineering to microfluidics [ 38 , 39 , 40 , 41 , 42 ].…”

“…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 ].…”