# Optimal Design of Electromagnetically Actuated MEMS Cantilevers

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**F**[6,7].

_{z}#### 2.1. Optimal Design of an Electromagnetically Actuated Cantilever: Direct Problem

**g**of the cantilever end, current

**I**, and magnetic induction

**B**, find:

- the stiffness k of the cantilever;
- the resonance frequency f of the cantilever;
- the force F
_{z}acting on the end region and its displacement Δz; - the electric resistance R (power-loss related) of the Lorentz loop.

_{1}is the cantilever length and L

_{2}is the tip length (see Figure 1).

^{−3}.

_{z}and its displacement Δz can be calculated as follows:

_{1}and the tip, with resistance R

_{2}):

^{4}Sm

^{−1}.

#### 2.2. Optimal Design of an Electromagnetically Actuated Cantilever: Inverse Problem

**g**= (g

_{1}, …, g

_{k}, …, g

_{n}) of geometric variables (e.g., for a polygonally-shaped end region, the coordinates of the relevant vertices), the inverse (or design) problem reads: given current I and magnetic induction B, find the shape

**g**= (g

_{1}, …, g

_{k}, …, g

_{n}) of the cantilever end region such that:

- the stiffness k(
**g**) of the cantilever is minimized; - the resonance frequency f(
**g**) is maximized; - the displacement Δz(
**g**) of the end region is maximized; - the electric resistance R(
**g**) of the Lorentz loop is minimized.

**g**), f(

**g**), Δz(

**g**), R(

**g**)] is originated. When more than one objective function is considered in the optimization, more solutions, belonging to the so-called Pareto front, are obtained. In particular, a solution is called Pareto optimal if there does not exist another solution that dominates it i.e., a solution that cannot be improved in any of the objectives without degrading at least one of the other objectives.

**g**is said to dominate another solution

_{1}**g**, if:

_{2}**g**is called Pareto indifferent with respect to a solution

_{1}**g**if:

_{2}_{1}= k(

**g**), f

_{2}= f(

**g**), f

_{3}= Δz (

**g**) and f

_{4}= R(

**g**). Our goal is, starting from a prototype geometry

**g**, to find a new geometry improving

_{0}**g**against the four objectives, according to Equations (6)–(9).

_{0}- w, arm width.
- L
_{1}, cantilever length. - L
_{2}, tip length. - b, cantilever width.

**g**;

**g**) is simply updated.

**d**(standard deviation) centered at the initial point

_{0}**m**(mean value);

_{0}**m**is externally provided, while

_{0}**d**is internally calculated on the basis of the bounds boxing the variation of the design variables.

_{0}**m**=

**m**and

_{0}**d**=

**d**, the generation of the design vector

_{0}**x**=

**m**+

**u d**then proceeds, resorting to a normal sample $\mathbf{u}\in (0,1)$. It is verified that

**x**fulfils bounds and constraints (i.e., that

**x**is feasible), otherwise a new design vector is generated until it falls inside the feasible region.

**x**) is then evaluated and the test if f(

**x**) dominates f(

**m**) (Equations (6) and (7)) is performed; if the test is successful,

**m**is replaced by

**x**(the so-called selection process), otherwise

**m**is retained.

#### 2.3. Fabrication Process

_{2}H

_{6}5% in N

_{2}at 270 °C (pre-diffusion) and high temperature annealing (diffusion of the boron dopant from previously deposited layer into silicon). The boron dopant profile can be optimized by controlling the duration time of pre-diffusion and duration time and temperature of the diffusion step.

^{20}cm

^{−3}.

## 3. Results

#### 3.1. Single-Objective Optimization Results

#### 3.2. Bi-Objective Optimization Results

#### 3.3. Tri-Objective Optimization Results

#### 3.4. Measurements on Optimal Cantilevers

- -
- two cantilevers corresponding to initial design,
- -
- final design, according to Opt2fz optimization result
- -
- final design, according to Opt3 optimization result

_{B}is the Boltzmann constant, T is the absolute temperature. By fitting parameters of Equation (11) to the measurements [18], the mechanical parameters are obtained. The thermomechanical noise formula is a result of the equipartition theorem [29]. Due to this theorem an object, which dissipates energy in thermal equilibrium, is subject to a fluctuation force. The measured resonance responses of the cantilevers shown in Figure 5a are shown in Figure 5b.

#### 3.5. Electromagnetic Actuation of the Cantilevers

## 4. Discussion

_{e}is the effective mass of the cantilever, m

_{g}is the mass of gold. The mass of the gold layer could be assessed from the volume.

_{1}, L

_{2}, b) and objective vector (k, f, Δz, R). This proves that a single solution simultaneously satisfying all the design criteria does not exist. From the optimization theory viewpoint, it is a design conflict problem that can be studied via Pareto optimality.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Design variables of the cantilever (black symbols) and schematic representation of the electromagnetic actuation by means of the Lorentz force (red lines).

**Figure 2.**The scanning electron microscope (SEM) image of the cantilever matrix after three technological steps (inter-operative control).

**Figure 3.**The scanning electron microscope (SEM) image of the final structure. Scanning parameters of the image. High Voltage (HV) = 10 kV; Working Distance (WD) = 5.7 mm; Horizontal Field of View (HFW) = 2.91 mm.

**Figure 4.**The high concentration of p dopant layer simulation result for the silicon on insulator (SOI) cantilever. Red and green curves represent two methods of doping concentration calculation.

**Figure 5.**(

**a**) example of manufactured cantilever array consisting of four cantilevers. Cantilevers 1 and 3 (from left to right) correspond to the initial design, while 2 and 4 are the optimal cantilevers after optimizations Opt2fz and Opt3. (

**b**) the resonance response of the manufactured cantilevers.

**Figure 6.**Results of the electromagnetic actuation of the optimal cantilevers (

**a**) results recorded in the low frequency region (100–1000 Hz) and (

**b**) in resonance. The experiments were performed under uniform 317 mT magnetic field.

w | L_{1} | L_{1} | b | |

Lower bound | 20 | 100 | 50 | 100 |

Upper bound | - | 600 | 100 | 150 |

w [µm] | L_{1} [µm] | L_{2} [µm] | b [µm] | k [Nm^{−1}] | f [kHz] | Δz [nm] | R [Ω] | |
---|---|---|---|---|---|---|---|---|

Initial | 20.0 | 500 | 50.0 | 100 | 4.32 × 10^{−2} | 8.00 | 925 | 470 |

Opt1k | 21.1 | 569 | 53.0 | 114 | 3.09 × 10^{−2} | 6.18 | 1482 | 512 |

Opt1f | 24.6 | 210 | 64.6 | 123 | 0.726 | 45.3 | 67.6 | 138 |

Opt1z | 22.0 | 568 | 63.9 | 128 | 3.25 × 10^{−2} | 6.21 | 1574 | 478 |

Opt1R | 47.6 | 211 | 80.7 | 119 | 1.39 | 45.2 | 34.3 | 69.3 |

w [µm] | L_{1} [µm] | L_{2} [µm] | b [µm] | k [Nm^{−1}] | f [kHz] | Δz [nm] | R [Ω] | |
---|---|---|---|---|---|---|---|---|

Initial | 20.0 | 500 | 50.0 | 100 | 4.32 × 10^{−2} | 8.00 | 925 | 470 |

Opt2kR | 24.1 | 557 | 62.3 | 110 | 3.76 × 10^{−2} | 6.45 | 1171 | 429 |

Opt2fz | 21.0 | 490 | 55.4 | 136 | 4.82 × 10^{−2} | 8.33 | 1129 | 438 |

Opt2fR | 60.9 | 210 | 86.1 | 135 | 1.79 | 45.4 | 30.3 | 56.4 |

Opt2zR | 27.2 | 569 | 57.7 | 111 | 3.99 × 10^{−2} | 6.18 | 1113 | 395 |

w [µm] | L_{1} [µm] | L_{2} [µm] | b [µm] | k [Nm^{−1}] | f [kHz] | Δz [nm] | R [Ω] | |
---|---|---|---|---|---|---|---|---|

Initial | 20.0 | 500 | 50.0 | 100 | 4.32 × 10^{−2} | 8.00 | 925 | 470 |

Final | 24.0 | 562 | 68.1 | 112 | 3.65 × 10^{−2} | 6.33 | 1226 | 429 |

Measured Quantities | Computed Quantities | |||||
---|---|---|---|---|---|---|

R [kΩ] | f [kHz] | Q | k [Nm^{−1}] | F_{min} [pN] | I_{min} [nA] | P_{min} [fW] |

1.89 | 6382 | 22.2 | 0.051 | 0.308 | 9.63 | 175 |

2.26 | 6386 | 22.3 | 0.053 | 0.313 | 9.79 | 217 |

1.88 | 6633 | 23.4 | 0.060 | 0.319 | 9.98 | 187 |

2.26 | 6634 | 23.4 | 0.055 | 0.306 | 9.55 | 206 |

1.88 | 6461 | 22.9 | 0.057 | 0.319 | 9.96 | 187 |

2.26 | 6502 | 22.9 | 0.056 | 0.315 | 9.84 | 219 |

1.88 | 6787 | 23.9 | 0.060 | 0.312 | 9.76 | 179 |

2.26 | 6799 | 24.6 | 0.064 | 0.318 | 9.93 | 223 |

Measured Quantities | Computed Quantities | ||||||
---|---|---|---|---|---|---|---|

R [kΩ] | f [kHz] | Q | k [Nm^{−1}] | F_{min} [pN] | I_{min} [nA] | P_{min} [fW] | |

Opt2fz, array1 | 2.16 | 5975 | 23.8 | 0.058 | 0.328 | 7.13 | 110 |

Opt2fz, array2 | 2.15 | 5819 | 25.3 | 0.063 | 0.336 | 7.30 | 115 |

Opt3, array1 | 1.74 | 5008 | 21.2 | 0.046 | 0.338 | 9.60 | 160 |

Opt3, array2 | 1.74 | 5205 | 22.3 | 0.038 | 0.294 | 8.35 | 121 |

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**MDPI and ACS Style**

Di Barba, P.; Gotszalk, T.; Majstrzyk, W.; Mognaschi, M.E.; Orłowska, K.; Wiak, S.; Sierakowski, A. Optimal Design of Electromagnetically Actuated MEMS Cantilevers. *Sensors* **2018**, *18*, 2533.
https://doi.org/10.3390/s18082533

**AMA Style**

Di Barba P, Gotszalk T, Majstrzyk W, Mognaschi ME, Orłowska K, Wiak S, Sierakowski A. Optimal Design of Electromagnetically Actuated MEMS Cantilevers. *Sensors*. 2018; 18(8):2533.
https://doi.org/10.3390/s18082533

**Chicago/Turabian Style**

Di Barba, Paolo, Teodor Gotszalk, Wojciech Majstrzyk, Maria Evelina Mognaschi, Karolina Orłowska, Sławomir Wiak, and Andrzej Sierakowski. 2018. "Optimal Design of Electromagnetically Actuated MEMS Cantilevers" *Sensors* 18, no. 8: 2533.
https://doi.org/10.3390/s18082533