# Large Stepwise Discrete Microsystem Displacements Based on Electrostatic Bending Plate Actuation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Concept

#### 2.1. Flexible Electrostatic Bending Plate Actuators

_{electrode}(Figure 1a). The rotor electrode is fixed to a spring-mounted rotor. The stator electrode is combined with a flexible cantilever. When a voltage is applied between the electrodes, the tips of the electrodes approach as soon as the electrostatic force between the electrodes exceeds the overall mechanical force of springs and electrodes. Therefore, the electrodes bend towards each other which is the origin to name these actuators bending plate actuators. The bending of the electrodes starts with the tips approaching each other as the electrodes feature the smallest stiffness at the tips. This behavior is examined experimentally in Section 4.2.2. With increasing voltage, the cantilever of the stator electrode bends down the distance b. The pull-in is completed as soon as the electrodes are completely in contact, as shown in Figure 1b.

_{a}. Thus, the flexibility of the cantilever reduces the pull-in voltage of the actuator as the stiffness of the electrodes increases with increasing distance to the tips. Therefore, the cantilever bending is a compromise between a reduction of the pull-in voltage and a reduction of the maximum displacement. In [19] we show that the maximum displacement range of the actuator x

_{a}is the electrode gap x

_{electrode}considering the bending b of the cantilever, which yields:

#### 2.2. System Function

_{1}, a

_{2}, …, a

_{j}) linked in a chain by connecting springs (k

_{1}, k

_{2}, …, k

_{j}) with the identical stiffness k. The first actuator a

_{1}has the smallest initial electrode gap x

_{1}. With each actuator, the initial gap increases, so the last actuator a

_{j}has the largest initial electrode gap x

_{j}. A main sinusoidal guiding spring k

_{g}[20] is directly connected to the last actuator and guarantees a pure translational system displacement x

_{system,i}. The connecting spring k

_{1}and the guiding spring k

_{g}are coupled to the substrate of the chip.

_{1}, this actuator displaces and its displacement is conducted through the system generating a system displacement x

_{system,1}. The displacement of the system x

_{system,i}when activating actuator a

_{i}is always identical to the reduction of x

_{j}of the last actuator a

_{j}(for i ≤ j) minus the deformation of the connecting springs. Therefore, considering the bending of the cantilever (Equation (1)), the maximum total system displacement x

_{system,j}is the electrode gap distance x

_{j}minus the bending of the cantilever b

_{j}.

#### 2.3. Step-by-Step and Collective Actuation

_{1}is activated, it travels the distance x

_{1}and the system travels the range x

_{system,1}. When the actuators a

_{1}and a

_{2}are activated, the system travels the range x

_{system,2}. By successively activating the individual electrostatic actuators, a stepwise system displacement is generated. Therefore, x

_{system,j}represents the displacement of the system composed of j actuators when all actuators are displaced and, e.g., x

_{system,3}represents the displacement of a system composed of j actuators when the actuators a

_{1}, a

_{2}and a

_{3}are displaced (for 3 ≤ j). The system displacement x

_{system,i}when activating i actuators in a system consisting of j actuators (i ≤ j) is described by Equation (2):

_{1,}this actuator generates a displacement x

_{1}that is conducted through the entire system. This initial displacement reduces the gaps between the electrodes of all following actuators a

_{2}to a

_{j}. For a structure composed of j actuators, Equation (3) describes the displacement x

_{i}of actuator a

_{i}when actuating all actuators with a smaller electrode gap distance (a

_{1}, a

_{2}, …, a

_{i}–

_{1}):

_{g}reduces the system displacement, whereas a higher stiffness of the connecting springs (k

_{1}, …, k

_{j}) increases the displacement.

#### 2.4. System Setup Based on Modelling and Simulation

#### 2.4.1. System Design Based on the Guiding Spring

_{system,i}and on Equation (3) to determine the electrode gaps x

_{i}. The mean and actual stiffness of the guiding springs are determined based on the COMSOL Multiphysics simulation results presented in Figure 4b.

_{1}has an electrode gap of 20 µm and actuator a

_{2}of 38.28 µm. When actuating a

_{1}, actuator a

_{1}will pull in and based on Equation (3) the actuator a

_{2}will travel the distance 18.28 µm, so that now the actuator a

_{2}has an electrode gap of 20 µm. For this setup, we assume that an equal electrode gap results in an equal voltage for complete pull-in and we neglect the cantilever bending.

#### 2.4.2. System Design for Different Step Numbers

_{system,j}is equal to the electrode gap of the last actuator a

_{j}minus the bending b

_{j}of the cantilever. Assuming a negligible small cantilever bending, the calculated total displacements of, the 5-, 8-, 10-, 13-, and 16-step systems amount to 54.7 µm, 67.9 µm, 74.5 µm, 82.5 µm and 88.9 µm, respectively. Consequently, the average step size decreases with an increasing number of steps. The reason is found in the increasing number of connecting springs that deform and therefore reduce the displacement that can be conducted to the following actuators. This deformation of the connecting springs is increased due to the high stiffness of guiding spring 1.

#### 2.4.3. Overview of the Modelled Systems

_{i}= 4.9 N/m at 20 µm displacement. The fabricated connecting springs are shown in Figure 3. They feature a length of 1965 µm, a thickness of 12 µm, a width of 50 µm and 1.5 sinusoids with an amplitude of 47.5 µm. The systems have 5, 8, 10, 13 and 16 steps. The 5-step systems can be activated step-by-step and collectively. Due to the high number of bond pads, systems with more than 5 steps can only be activated collectively.

## 3. Fabrication

_{2}layer (Figure 7c) is used as a hard mask for the patterning of the systems. Then, also the device layer is deep etched (Figure 7d). The systems are released by HF-vapor etching (Figure 7e). The electrical isolation of the electrodes is performed by depositing 400 nm silicon nitride (SiN) layer with a low-stress PECVD process (Figure 7f). The low-stress process is required to overcome residual deflections of the very slim electrodes. Single chips are placed on a carrier wafer coated with 50 nm aluminum and flipped to minimize the spacing of the electrodes and the carrying wafer for preventing a parasitic coating of the bond-pads. Finally, the chips are assembled on a printed circuit board (PCB) and wire bonded (Figure 3).

## 4. System Characterization

#### 4.1. Experiment and Characterization Setup

#### 4.2. System Characterization Results

#### 4.2.1. Overview of Results

#### 4.2.2. Design-Based Electrostatic Bending Plate Actuator Behavior

#### 4.2.3. Step-by-Step Actuation of the 5-Step Systems

_{system,5}, depending on the activated steps. The system shows an approximately linear displacement. A comparison with the calculated displacement shows that the analytical model and the experimental results fit very well. The experimentally derived cantilever bending b differs from step to step. For step 1 it amounts to 11.5 µm, for steps 2 to 5 the cantilever bending amounts to 10.1 µm, 10.1 µm, 15.8 µm and 11.0 µm.

#### 4.2.4. Comparison of Step-by-Step and Collective Actuation

#### 4.2.5. Characterization of Microsystems with 5 to 16 Steps

_{1}to a

_{4}pull-in in turn with increasing voltage. However, the actuators a

_{5}to a

_{16}pull-in at the same voltage (Figure 14b and Figure 15c). This is attributed to the low stiffness of the guiding spring supporting to conduct an impulse of movement to the following actuators and resulting in a collective pull-in. Furthermore, the high number of interacting electrostatic actuators increases the electrostatic force in the system which is highly facilitating a pull-in.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Setup of the bending plate actuator, (

**b**) completely pulled-in bending plate actuator, (drawings not to scale).

**Figure 3.**(

**a**) 5-step system (system 2) for step-by-step and collective actuation, (

**b**) 16-step system (system 6) for collective actuation, (stacked device photos).

**Figure 4.**(

**a**) Geometry of the guiding springs 1 and 2; (

**b**) stiffness of guiding springs 1 and 2 based on solid state COMSOL Multiphysics simulation.

**Figure 5.**System displacement and electrode gap distance of 16-step systems depending on mean and actual stiffness value design guided by (

**a**) guiding spring 1 (actual stiffness value: system 7 presented in Table 1), (

**b**) guiding spring 2 (mean stiffness value: system 6 presented in Table 1). The values are obtained analytically (Equations (2) and (3)) by calculating with the spring stiffness values that are obtained by COMSOL simulation. The electrode gap amounts to 20 µm for each actuator.

**Figure 6.**Analytically derived electrode gaps for systems with a mean stiffness value design featuring (

**a**) guiding spring 1, (

**b**) guiding spring 2.

**Figure 7.**Fabrication process (

**a**) etching of bond pads, (

**b**) deep-etching of handle layer, (

**c**) PECVD of 400 nm SiO

_{2}, (

**d**) deep-etching of device layer, (

**e**) HF-vapor etching, (

**f**) deposition of 400 nm SiN, (drawings not to scale).

**Figure 9.**Pull-in depending on electrode thickness, (

**a**–

**d**) c = d = 5 µm, (

**e**–

**h**) c = 10 µm, d = 5 µm, detailed description in the continuous text.

**Figure 10.**Displacement of the bending plate actuator and bending of the cantilever depending on the applied voltage, for the actuator of step 5 of system 1.

**Figure 11.**Step-by-step actuation of system 2 (left hand side) at 97 V, relays closed at (

**a**) step 1, (

**b**) steps 1 to 3, (

**c**) experimental and analytical displacement of system 2 at 97 V control voltage, Table 2 presents the exact measured results.

**Figure 12.**Time-dependent displacement of the single steps at step-by-step activation of (

**a**) system 1, (

**b**) system 2.

**Figure 13.**Collective actuation of system 2 at (

**a**) 44 V, (

**b**) 50 V, (

**c**) experimental displacement of the single steps of system 2, the voltage is increased in 1 V-steps every 2 s.

**Figure 14.**System displacement depending on applied voltage of the 5-, 8-, 10-, 13- and 16-step systems during collective activation, comparison of (

**a**) systems 2 to 6, (

**b**) system 6 and system 7, the voltage is increased in 1 V-steps every 2 s.

**Figure 15.**Voltage-dependent total displacement of the 16-step system 6, at (

**a**) 0 V, (

**b**) 53 V, and (

**c**) 54 V and of the 16-step system 7 at (

**d**) 0 V, (

**e**) 53 V and (

**f**) 118 V.

**Table 1.**Modeled systems, the simulated mean stiffness of guiding spring 1 is k

_{g}= 2.9 N/m and the simulated stiffness of the the connecting springs is k

_{i}= 4.9 N/m.

System | Number of Steps | Activation | Design/ Stiffness Value | Guiding Spring | Max. Calculated Displacement | Chip Size [µm × µm] | Presented in Figure |
---|---|---|---|---|---|---|---|

1 | 5 | step-by-step and collective | mean | 1 | 80 µm | 6945 × 10,253 | 6a |

2 | 5 | step-by-step and collective | actual | 2 | 94 µm | 6945 × 10,253 | 3a |

3 | 8 | collective | mean | 2 | 143 µm | 8865 × 9157 | 6b |

4 | 10 | collective | mean | 2 | 171 µm | 10,255 × 9157 | 6b |

5 | 13 | collective | mean | 2 | 212 µm | 12,392 × 9157 | 6b |

6 | 16 | collective | mean | 2 | 242 µm | 14,407 × 9157 | 3b, 5b, 6b |

7 | 16 | collective | actual | 1 | 156 µm | 14,407 × 9157 | 5a |

System | Max. Experimental Displacement | Voltage for Max. Displacement | System Properties |
---|---|---|---|

1 | x_{system,5} = 71.9 ± 0.5 µm | 82 V (step-by-step) 71 V (collective) | x_{system,1} = 5.9 µm, x_{system,2} = 22.0 µm, x _{system,3} = 38.5 µm, x_{system,4} = 55.4 µm, oscillation during step-by-step actuation |

2 | x_{system,5} = 89.4 ± 0.8 µm | 97 V (step-by-step) 69 V (collective) | x_{system,1} = 10.6 µm, x_{system,2} = 27.9 µm, x _{system,3} = 47.3 µm, x_{system,4} = 61.4 µm, oscillation during step-by-step actuation |

3 | x_{system,8} = 138.4 ± 2.3 µm | 74 V | pull-in order: a_{1} + a_{2}, a_{3} + a_{4}, a_{5}, a_{6}, a_{7}, a_{8} |

4 | x_{system,10} = 168.3 ± 3.1 µm | 65 V | pull-in order: a_{1}, a_{2}–a_{7}, a_{8}, a_{9}, a_{10} |

5 | x_{system,13} = 202.7 ± 3.6 µm | 68 V | pull-in order: a_{1}, a_{2}–a_{4}, a_{5}–a_{8}, a_{9}, a_{10}–a_{12}, a_{13} |

6 | x_{system,16} = 230.7 ± 0.9 µm | 54 V | pull-in order: a_{1}, a_{2}, a_{3}, a_{4}, a_{5}–a_{16} |

7 | x_{system,16} = 138.9 ± 1.7 µm | 118 V | pull-in order: a_{1}, a_{2} + a_{3}, a_{4}–a_{6}, a_{7}, a_{8}, a_{9}–a_{12}, a_{13,} a_{14}, a_{15}, a_{16} |

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**MDPI and ACS Style**

Schmitt, L.; Hoffmann, M.
Large Stepwise Discrete Microsystem Displacements Based on Electrostatic Bending Plate Actuation. *Actuators* **2021**, *10*, 272.
https://doi.org/10.3390/act10100272

**AMA Style**

Schmitt L, Hoffmann M.
Large Stepwise Discrete Microsystem Displacements Based on Electrostatic Bending Plate Actuation. *Actuators*. 2021; 10(10):272.
https://doi.org/10.3390/act10100272

**Chicago/Turabian Style**

Schmitt, Lisa, and Martin Hoffmann.
2021. "Large Stepwise Discrete Microsystem Displacements Based on Electrostatic Bending Plate Actuation" *Actuators* 10, no. 10: 272.
https://doi.org/10.3390/act10100272