# Design Guideline for a Cantilever-Type MEMS Switch with High Contact Force

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{eff}. The contact resistance R

_{c}is determined as follows [27]:

_{eff}is written as:

_{c}is the contact force, δ is the asperity peak radius of curvature, and E

_{eff}is the effective Hertzian modulus derived from:

_{1}, E

_{2}and υ

_{1}, υ

_{2}are the Young’s modules and Poisson’s coefficients of the materials one and two. When the force exceeds 200 µN, plastic deformation takes place, and the effective radius is determined as:

_{c}depends on the switch design and can vary over a wide range. For the elastic and plastic regimes, R

_{c}is inversely proportional to F

_{c}

^{1/3}and F

_{c}

^{1/2}, respectively. Thus, one has to increase the force in order to reduce the resistance. In addition, a large F

_{c}helps to break contaminating films and enhances the stability of R

_{c}from cycle to cycle [35]. Experimental data suggest the minimum required value of 100 µN per contact [29,30,31,33,38].

_{c}expands the contact area and leads to higher adhesion. Therefore, the growth of the contact force must be accompanied by an increment in the restoring force F

_{r}in order to ensure de-actuation. It is generally accepted that F

_{r}of more than one-third of F

_{c}is necessary for a stable restoration action [44]. This relation has to be considered when the switch is designed.

_{e}= 20 µm. The driving electrode surrounds the signal one and provides the overlap area A = 430 µm

^{2}with the cantilever. The widening and surrounding are commonly used to expand the electrostatic field area without significant increase in the switch size [46,49,50]. The cantilever has the thickness t = 2.0 µm. A contact bump with the height h = 0.5 µm is located on its bottom surface. The air gap between the cantilever and electrodes is g

_{0}= 1.5 µm. A detailed description of the switch is given in [51].

_{0}− h. In the bottom position, its profile can be approximated by a straight line, as shown in Figure 3. In this case, the gap between the cantilever and the driving electrode linearly depends on the longitudinal coordinate:

_{1}= 25 µm and x

_{2}= 50 µm are the coordinates of the left and right electrode edges. This gap is used to calculate the electrostatic force in the closed state:

_{0}is the electric constant, and V is the voltage applied to the driving electrode with respect to the grounded cantilever. Electrostatic switches typically operate at the driving voltage of several tens of volts. At V = 90 V, the electrostatic force is equal to F

_{es}= 28 µN.

_{r}= 18 µN.

_{c}= 10 µN, which is a rather low value. With such a force, R

_{c}is large and varies dramatically from cycle to cycle, as we demonstrated previously [54,55]. Increasing F

_{c}by reducing F

_{r}is limited and raises the tendency to stiction. Another way is to raise F

_{es}by using higher V, but higher voltage increases power consumption and may cause the collapse of the cantilever. Enlarging A is also unacceptable, since the switch loses the advantages of a miniature device. Thus, the force is enhanced by manipulating the vertical dimensions h, g

_{0}and t.

_{c}from 0.6 to 30 µN [15,16,46,47,48]. These devices have various vertical dimensions. There are no specific values that can be considered conventional. For example, a switch with h = 1 μm, g

_{0}= 2.5 μm and t = 2 μm provides the contact force of 18 μN [15]. However, some cantilever-based switches with similar vertical size provide a significantly higher force of 113–301 µN [50,56,57]. They use large cantilevers with the length of 300–485 μm, which suffer from bending under residual mechanical stress, reduce switching speed and increase parasitic capacitance.

_{pull-in}, which is calculated using a well-known expression [52]:

_{pull-in}, the cantilever excessively deforms after actuation and touches the driving electrode that results in the switch failure due to a short circuit. This phenomenon is called secondary pull-in or collapse. The collapse voltage V

_{collapse}is determined by FEM simulation with commonly used software. For reliable operation of the switch, V

_{collapse}must significantly exceed V

_{pull-in}.

## 3. Results and Discussion

#### 3.1. Choosing the Vertical Dimensions

_{c}on h is shown in Figure 4a. Decreasing the height from 0.5 to 0.1 µm increases the contact force from 10 to 50 µN. The growth is explained by an increase in the electrostatic force due to a drop of the distance between the cantilever and the electrode in the closed state. For h from 0.2 to 0.5 µm, the analytical results agree with FEM predictions. However, for h = 0.1 µm, the simulation provides F

_{c}= 89 µN, which is almost two times higher in comparison with analytics. This discrepancy is determined by buckling of the cantilever toward the electrode, which is not taken into account in the analytical model. Decreasing h linearly increases the restoring force from 18 to 26 µN, as shown in Figure 4a.

_{collapse}= 220 V. Therefore, the range from 0.2 to 0.5 µm ensures safe operation of the switch. For this range, the strongest forces F

_{c}= 32 µN and F

_{r}= 24 µN are achieved at h = 0.2 µm, so this value is considered as an optimal height, which is used in further calculations.

_{c}and F

_{r}on t for various values of g

_{0}is demonstrated in Figure 5a. For the initial thickness t = 2.0 µm, decreasing the gap from 1.5 to 0.6 µm raises the contact force from 32 to 164 µN, but the restoring force drops from 24 to 7 µN. This drop has to be compensated for stable overcoming of stiction. The compensation is easily achieved by increasing the cantilever thickness, because k~t

^{3}. But the growth of the restoring force reduces the contact force, according to Equation (11). Therefore, one has to choose the values of g

_{0}and t, which satisfy the conditions F

_{c}≥ 100 µN and F

_{r}≥ F

_{c}/3. The first condition is possible only for the gap of 0.6 and 0.8 µm, as shown in Figure 5a. The second relation is valid for the values of t located to the right of the vertical lines marked on this graph. For g

_{0}= 0.8 µm, the minimal thickness is of 2.9 µm. But in this case F

_{c}= 92 µm, so the first condition is not satisfied. For g

_{0}= 0.6 µm, the minimal thickness is equal to 3.6 µm. The forces are F

_{c}= 128 µN and F

_{r}= 43 µN, so both conditions are fulfilled, and this gap is suitable. Nevertheless, the thickness can be slightly increased in order to raise the restoring force and ensure the margin of reliability. For t = 4.0 µm, the switch develops contact and restoring forces of 112 and 59 µN, respectively. FEM simulation provides F

_{c}= 109 µN and confirms the validity of the analytical model.

_{0}significantly lowers V

_{pull-in}, while increasing t raises V

_{pull-in}due to increasing k. The initial switch with g

_{0}= 1.5 µm and t = 2.0 µm demonstrates V

_{pull-in}= 70 V, while the optimized device with g

_{0}= 0.6 µm and t = 4.0 µm is actuated at V

_{pull-in}= 50 V. Thus, manipulation of the vertical dimensions enhances the forces without increasing the operating voltage. It is worth noting that decreasing g

_{0}from 1.5 to 0.6 µm and increasing t from 2.0 to 4.0 µm slightly lowers the collapse voltage from 220 to 200 V. However, V

_{collapse}still significantly exceeds the driving voltage and ensures safe operation.

_{c}from 10 to 112 µN, i.e., by more than an order of magnitude. With such a force, an elastic deformation of the contact material takes place. According to Equations (1) and (2), the growth of F

_{c}should reduce the contact resistance by 2.2 times. However, this assumption is valid for clean surfaces. For real contacts, exceeding the threshold of 100 µN allows one to expect a more significant decrease in R

_{c}[29,30,31,33,38] and its stabilization due to the breaking of contaminating films. The restoring force grows from 18 to 59 µN and is approximately half of F

_{c}, which ensures a reliable overcoming of stiction. An additional benefit is the reduction in V

_{pull-in}from 70 to 50 V. These results are achieved by reducing the bump height from 0.5 to 0.2 µm and the gap from 1.5 to 0.6 µm as well as by increasing the cantilever thickness from 2.0 to 4.0 µm. The switch design is not modified principally, and the lateral size remains compact. The switch can be fabricated with the technical process established by authors [58]. The route does not require any changes except the time for deposition and etching of structural materials.

#### 3.2. Double Cantilever Design

_{e}. It has two fixed regions and two bumps, which increase stability in the bottom position. The force per bump remains unchanged and equal to 112 µN, while the total contact force F

_{c,total}= 2F

_{c}= 224 µN is doubled. The restoring force F

_{r,total}= 2F

_{r}= 118 µN also grows two times compared to a single structure, because combining two cantilevers doubles the stiffness. The pull-in voltage does not change, since the growth of k is compensated by the enlargement of A. It is important to note that the double switch should have two times lower contact resistance due to doubling the contact area. The dual design was first introduced by Northeastern University [59] and elaborated in several papers [47,48,49]. Thanks to high reliability, it was transformed by Radant MEMS to one of the most successful switches on the MEMS market [60]. However, the underlying reasons for choosing the cantilever shape and size were not explained.

#### 3.3. Multiple Cantilever Design

_{c,bump}on n is shown in Figure 8. The number of beams varies from 1 to 9, while the number of contacts takes the value from 1 to 5, according to the rule (n +1)/2. The switch with n = 3 develops F

_{c,bump}= 168 µN, which is 1.5 times higher compared to the basic design with n = 1. Increasing the number of cantilevers to n = 9 raises the specific force to 202 µN. But it should be noted that the growth of F

_{c,bump}slows down with increasing n. The cantilever with the large number of bumps may provide unequal force distribution among contacts due to the technological variation in the bump height. In addition, using multiple cantilevers increases the lateral size of the switch. An optimal case is the triple cantilever with two bumps, as schematically illustrated in Figure 9. It provides a uniform distribution of forces and has the reasonable width of 60 µm.

_{r}with increasing n. The dependence of F

_{r,bump}on n is depicted in Figure 8. As for the contact force, the growth of the restoring force slows down with increasing the number of beams. The largest increment takes place for the transition from n = 1 to n = 3. The optimal switch with the triple cantilever provides F

_{r,bump}= 88 µN.

_{collapse}by varying the distance between the bumps w

_{b}indicated in Figure 9. The dependence of V

_{collapse}on w

_{b}is shown in Figure 10a. Initially, the distance is equal to 40 µm, and the collapse takes place at V = 130 V. The deformation of the cantilever is concentrated between the contacts, as demonstrated in Figure 10b. Reducing w

_{b}to 34 µm lowers the deformation of the central part and raises V

_{collapse}to 200 V. Further reducing the distance drops the collapse voltage again due to the excessive bending of the cantilever edges. Thus, w

_{b}= 34 µm is an optimal value, which provides the same collapse voltage as for the basic single-beam design.

_{c,bump}on n is a horizontal line shown in Figure 8 by green color. The switch with four cantilevers and two contacts develops a 30% higher specific force than the triple cantilever with two bumps. However, it has the collapse voltage of 90 V or lower, depending on the position of bumps. This value is close to the pull-in voltage, and safe operation of the switch is not ensured. This situation takes place for all the switches designed according to the rule n/2. Therefore, the rule (n +1)/2 is optimal.

_{c}is highlighted in the papers [39,40,41,42,43,44,45,50,56,57]. However, in these works, the force is increased by enlarging lateral dimensions of the electrodes. In particular, the movable plate of the switch from [39] has a size of about 500 µm. The innovation of the present work is that it describes the way to increase F

_{c}of a compact switch. A device with a 50 µm long cantilever and a specific contact force of more than 150 µN has not been described previously. The parallel connection of several cantilevers and choosing the number and position of contact bumps is also proposed for the first time.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**The dependence of the contact and restoring force (

**a**) and collapse voltage (

**b**) on the bump height. Solid and dashed lines at the image (

**a**) correspond to contact and restoring force, while red and blue colors indicate analytical and FEM calculations. The inset at the image (

**b**) illustrates the deformation of the collapsed cantilever.

**Figure 5.**The dependence of the contact and restoring forces (

**a**) and pull-in voltage (

**b**) on the cantilever thickness for various values of the gap indicated by colors. Vertical lines at the image (

**a**) mark the thickness, for which F

_{r}is equal to F

_{c}/3. Solid and dashed lines correspond to contact and restoring force, while different colors correspond to various values of the gap, as shown at the image (

**b**).

**Figure 6.**The switch with two contact bumps located on a single cantilever (

**a**) and on a double cantilever (

**b**), top view.

**Figure 8.**The dependence of the contact and restoring force per bump on the number of cantilevers and bumps.

**Figure 10.**(

**a**) The dependence of the collapse voltage on the distance between contacts shown by red line. The upper inset shows the switch with w

_{b}= 40 µm, while the bottom one corresponds to w

_{b}= 30 µm, as indicated by arrows. (

**b**) Profiles of the free end of the cantilever for the different values of w

_{b}. The driving voltage is equal to 90 V. Vertical marks indicate the position of the bumps. The inset illustrates deformation of the cantilever with w

_{b}= 40 µm.

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**MDPI and ACS Style**

Uvarov, I.V.; Belozerov, I.A.
Design Guideline for a Cantilever-Type MEMS Switch with High Contact Force. *Micro* **2024**, *4*, 1-13.
https://doi.org/10.3390/micro4010001

**AMA Style**

Uvarov IV, Belozerov IA.
Design Guideline for a Cantilever-Type MEMS Switch with High Contact Force. *Micro*. 2024; 4(1):1-13.
https://doi.org/10.3390/micro4010001

**Chicago/Turabian Style**

Uvarov, Ilia V., and Igor A. Belozerov.
2024. "Design Guideline for a Cantilever-Type MEMS Switch with High Contact Force" *Micro* 4, no. 1: 1-13.
https://doi.org/10.3390/micro4010001