# Casimir Effect in MEMS: Materials, Geometries, and Metrologies—A Review

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

_{c}, depends only on the area of the plates A and their separation distance d: F

_{c}= π h c/480 A/d

^{4}[1], where h and c denote the Planck constant and light velocity, respectively

_{i}is smaller than the force outside F

_{o}. The pressure (force per area) from outside is higher than from inside, generating an attraction between the two plates.

## 2. Survey about Experimental and Theoretical Studies of Casimir Forces

#### 2.1. Plate–Plate Geometries with Different Material Combinations

#### 2.2. Lens/Sphere–Plate and Sphere–Sphere Geometries with Different Material Combinations

**Figure 3.**Studies of Casimir forces on spheres/lens–plates and sphere–sphere geometries. (

**a**) Quartz lens–plate [43,44], (

**b**) silica lens–plate [45,46], (

**c**) coated Cr lens–plate [47], (

**d**) coated Cu and Au sphere–plate [48], (

**e**) coated Al sphere–plate [49,50], (

**f**) coated Au sphere–plate [51,52,53], (

**g**) conducting hemispheres [57], (

**h**) goad-coated spheres [59], (

**i**) concentric spheres [58].

#### 2.3. Cylinder–Cylinder/Plate/Sphere Geometries with Different Material Combinations

#### 2.4. Special Geometries with Different Material Combinations

## 3. Main Measurement Methods: Review of Casimir Metrology

_{3}N

_{4}nanomembrane as a pressure sensor, which was under the gold plate (in Figure 6i). Using a fibre interferometer to measure the nanomembrane displacement, the Casimir force could be more precisely detected [90].

## 4. Influence of Further Parameters on Casimir Forces

#### 4.1. Influence of Thermal Effects

#### 4.2. Influence of Conductivity Effects

#### 4.3. Influence of Surface Roughness

## 5. Influences of Casimir Forces on the Characteristics of MEMS/NEMS Devices

#### 5.1. Influences of Casimir Forces on Stiction and Adhesion

#### 5.2. Influences of Casimir Forces on Pull-in Voltages

#### 5.3. Influences of Casimir Forces on Vibrational Properties

#### 5.4. Influences of Casimir Forces on Chaotic Motion

## 6. Applications of Casimir Forces in MEMS/NEMS Field

#### 6.1. Highly Sensitive Sensors

#### 6.2. Non-Contact Actuators

#### 6.3. Stiction-Related Applications

#### 6.4. Heat Transfer Devices and Actuators

#### 6.5. Optical Applications

#### 6.6. Harvesting Devices

#### 6.7. Applications in Quantum Computation and Communication

#### 6.8. The Role of Machine Learning

## 7. Self-Assembling Structures

#### Three-Dimensional Self-Assembly into Yin–Yang Structures

^{−6}N is required to counteract the restoring elastic forces for an area of 5600 µm

^{2}, implying a sufficient gap distance of less than 30 nm, see Figure 8. Further calculations adjusting for realistic material properties and non-ideal conditions confirm that the actual gap distance of 10 nm produces a much larger Casimir force, indicating the shutters are pressed together more than ten times stronger than the minimum required force. This force density significantly increases as the gap distance decreases, affirming the tight attachment of the shutters through Casimir forces, which are quantitatively consistent with the observed stability and configuration of the paired shutters.

- Dimensions A = 400 µm × 14 µm = 5600 µm
^{2}. - Mass per unit area: ${\sigma}_{m}$ = 0.62 g/m
^{2}considering the Al-Cr-Al metal stack. - potential difference: $U=1$ mV (due to charge fluctuations based on deviations in the working function of the facing aluminium surfaces).
- Contact angle: $\theta $ ≈ 20° (for isopropanol on hydrophilic aluminium).
- Surface tension: ${\gamma}_{l}$ = 17 mN/m (for isopropanol at 68.5 °C) [176].
- Hamaker constant: $H=4.554\times {10}^{-19}$ (for aluminium) [64].

- 1.
- Gravitational force per unit area:

^{3}/(kg s) is the gravitational constant.

- 2.
- Electrostatic force per unit area:

_{0}= 8.854 × 10

^{−12}F/m is the vacuum permittivity, and U is the potential difference.

- 3.
- Capillary force per unit area:

- 4.
- Casimir force per unit area in retarded limit (original formula of H. B. G. Casimir):

- 5.
- Casimir force per unit area in non-retarded limit (based on VdW interaction):

_{Al2}). Paired shutters form Yin–Yang shapes under Casimir forces, with geometries characterised by varying eccentricity (0.36 < ε < 0.67) and overlapping lengths (l

_{o}ranging 8–31 µm) as shown in Figure 10. Three pairing scenarios are identified based on SEM and CLSM imaging, revealing nonlinear dynamics in the assembly process. The fabrication process involves layer deposition and sacrificial layer removal, enabling controlled assembly [42].

## 8. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic illustration of Casimir forces between parallel plates using the quantum electromagnetic model. F

_{0}represents the forces exerted on the plates due to the quantum waves outside the plates, and F

_{i}refers to the one in between the plates.

**Figure 4.**Studies of Casimir forces on cylinders with different geometries. (

**a**) Crossed cylinders of mica and silica [3,45,60], (

**b**) crossed cylinders of gold [61], (

**c**) perfectly conducting, parallel cylinders [62,63], (

**d**) cylindrically bent metallic blades [64,65,66], (

**e**) cylinder–plate of perfect metals [67,68], (

**f**) perfectly conducting cylinder–sphere [69] (completely redrawn by the ideas of these references).

**Figure 5.**Different geometries of theoretical studies about Casimir forces. (

**a**) Perfectly conducting and dielectric wedges [70,71,72], (

**b**) corrugated plates [73,74], (

**c**) corrugated sphere–plate [75,76], (

**d**) squares between two walls [77], (

**e**) parallel metal plates with interleaved brackets [78], (

**f**) metal particles above plate with a hole [79], (

**g**) sphere–plate immersed in liquid [55,56,80,81], (

**h**) silicon plate with trench arrays and gold sphere [82,83], (

**i**) parallel plates with protrusions [84,85] (completely redrawn by the ideas of these references). The following abbreviations are used: attractive force (A), repulsive force (R) and a force of zero (Z).

**Figure 6.**Main experiments of Casimir force. (

**a**) Leverage system [43,44], (

**b**) balanced levers system [33], (

**c**) double cantilever spring system [3], (

**d**) torsion pendulum system [48], (

**e**) AFM system for plate–sphere [49,50,51,55,56,86,87], (

**f**) AFM system for corrugated plate–sphere [75,76,88], (

**g**) micromachined torsional devices [42,52,82,89], (

**h**) fibre interferometer–cantilever system [41], (

**i**) fibre interferometer–nanomembrane system [90], (

**j**) piezoelectric tube–bimorph cantilever system [61], (

**k**) vibrating plate system [91,92], (

**l**) comb and amplifier system [84,85] (completely redrawn by the ideas of these references).

**Figure 8.**

**Left**: Focused ion beam micrograph of the area where two microshutters come close together with a gap around 15 nm.

**Top right**: Comsol simulation of the steps to estimate the distance d. (

**a**) Un-actuated shutter, (

**b**) un-actuated shutter (dotted) and shutter (full line) actuated via an external force F

_{ext}acting on the area A (its cross-section highlighted as a red stripe), (

**c**) elastic force F

_{elast}and counteracting external force F

_{ext}on area A, (

**d**) the identical force equilibrium with the same but shifted forces, (

**e**) both shutters in grey overlapping within A (red), (

**f**) force equilibrium for the right shutter: restoring elastic force Felast and counteracting Casimir force F

_{C}, acting on the right area A (red), (

**g**) force equilibrium also involving forces acting on the left shutter and formation of a plate capacitor arrangement (red) with known area A and distance d to be determined.

**Bottom right**: Model calculations of the obtained Casimir force densities depending on the distance between the shutter blades d for (1) the Casimir approach (red line), (2) the Hamaker approach (dashed blue line) and (3) the exact model (solid light blue line), respectively. Modified from [64].

**Figure 9.**Graph showing comparison between different forces including gravitational, electrostatic, capillary and Casimir (retarded and non retarded) forces based on the introduced model. On the horizontal axis is the separation distance between the metal plates and on the vertical axis is the force density. The black line represents the gravitational forces, the red line represents the electrostatic forces, the blue line represents the capillary forces, the green line represents Casimir forces in the retarded regime and the magenta is the Casimir forces in the nonretarded regime.

**Figure 10.**Paired shutter arrangement as checkerboard (

**a**) and tubes (

**b**). The pairing and the overlapping area between the shutter blades, A and B, and the fitted orange and blue ellipses to identify both of the shutter blades. The extracted fit parameters are the major and minor axes (

**c**).

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**MDPI and ACS Style**

Elsaka, B.; Yang, X.; Kästner, P.; Dingel, K.; Sick, B.; Lehmann, P.; Buhmann, S.Y.; Hillmer, H.
Casimir Effect in MEMS: Materials, Geometries, and Metrologies—A Review. *Materials* **2024**, *17*, 3393.
https://doi.org/10.3390/ma17143393

**AMA Style**

Elsaka B, Yang X, Kästner P, Dingel K, Sick B, Lehmann P, Buhmann SY, Hillmer H.
Casimir Effect in MEMS: Materials, Geometries, and Metrologies—A Review. *Materials*. 2024; 17(14):3393.
https://doi.org/10.3390/ma17143393

**Chicago/Turabian Style**

Elsaka, Basma, Xiaohui Yang, Philipp Kästner, Kristina Dingel, Bernhard Sick, Peter Lehmann, Stefan Yoshi Buhmann, and Hartmut Hillmer.
2024. "Casimir Effect in MEMS: Materials, Geometries, and Metrologies—A Review" *Materials* 17, no. 14: 3393.
https://doi.org/10.3390/ma17143393