# The Mechanism of Layer Stacked Clamping (LSC) for Polishing Ultra-Thin Sapphire Wafer

^{*}

## Abstract

**:**

_{a}) of 1.52 nm and a flatness (PV) of 0.968 μm was obtained.

## 1. Introduction

## 2. Principle of Layer Stacked Clamping (LSC)

## 3. Adhesion Mechanism of LSC

#### 3.1. Fractal Theory of Rough Surface

^{n}determines the frequency spectrum of the surface roughness, the lowest frequency is related to the length L of the sample as γ

^{nl}= 1/L.

_{pe}of the micro-bulge when it changes from elastic deformation to plastic deformation (Equation (4)):

_{b}is the hardness of material, E* is Elastic Modulus, ${E}^{*}=\left(\right(1-{v}_{\mathrm{A}}^{2})/{E}_{\mathrm{B}}+(1-{v}_{\mathrm{B}}^{2})/{E}_{\mathrm{B}}{)}^{2}$, v

_{A}, v

_{B}are the Poisson’s ratio of surface A and surface B, respectively, E

_{A}, E

_{B}are the elastic modulus of surface A and surface B, respectively.

_{pe}can be obtained as:

#### 3.2. Van der Waals Force Adhesion Model

_{L}is the height of the micro-bulge, and δ

_{l}is the maximum deformation amount of the elastoplastic deformation of the micro-bulge.

_{n}, its value is δ

_{n}= δ

_{h}+ δ, and the area of the contact area is a, then the van der Waals work received by the micro projection which is based on the model of Israelachvili [35] is:

_{AWB}, J

_{AB}are the Hamaker constant. As shown in Equations (8) and (9) [35]:

_{A}, J

_{W}, and J

_{B}are the Hamaker constants of surface A, liquid, and surface B, respectively.

_{n}, and the height from the surface B to the top of the micro-bulge is h + δ

_{h}− δ

_{n}, the research of Israelachvili gave the van der Waals force model between the ball and the plane [35], which can be obtained by Equation (11):

_{L}is the bottom area of the largest micro-bulge, and ψ is the extended domain factor of the distribution of micro-bulge, and its expression is:

_{a}, and its expression is shown in Equation (14). The real contact area of the deformed micro-bulge is A

_{r}, and the van der Waals force in this area shows the interaction of two planes, the expression is shown in Equation (15):

_{r}to the apparent area A

_{a}is:

_{a}, the van der Waals work between surface A and surface B is:

_{h}is the bottom area of the largest micro-bulge which is not in contact with surface B. Define k is the ratio of area a

_{l}to a

_{L}, as k = a

_{l}/a

_{L}, therefore, the value of the critical contact area ratio k

_{pe}for different materials can be obtained by Equation (18):

_{e}shows as Equation (20), and the expression of a

_{h}shows in Equation (21):

#### 3.3. Capillary Adhesion Force

_{s}is the surface tension of the liquid in air, for water, γ

_{s}= 72 × 10

^{−3}N/m; r

_{o}is the narrowest “neck” radius of the liquid bridge meniscus, and θ

_{A}and θ

_{B}are contact angle of non-flat rough surface A and surface B, respectively; φ is the angle between the surface tangent and the horizontal plane; P

_{c}is the Laplace’s equation, and its expression is shown in Equation (27), which is related to the liquid surface tension γ

_{s}and the radius of meniscus curvature r:

_{o}of the curved surface is shown as:

_{B}is determined, the volume of the droplet determines the height H’ of the curved surface when the maximum capillary force is generated. Assume V is the droplet volume, Equation (31) shows the relation between V and H’:

_{c}, then the value of s is shown in Equation (33):

## 4. Adhesion Force Experiment and Discussion of LSC

#### 4.1. Experiment Preparation

_{a}, root mean square roughness (RMS), sampling frequency ω and other related parameters was obtained by Taylor Hobson’s surface profiler. Table 1 shows the parameters of different materials and roughness. Five different surface roughness of 304 stainless steel were measured to study the effect of roughness on the interface force.

_{a}[38]:

_{l}is the cutoff frequency and ω

_{h}is the high frequency which is determined by instrument resolution and filtering:

#### 4.2. Results and Discussion

^{−4}N. When the surface roughness is decreased, the micro-bulge on the rough surface are smaller, which increases the van der Waals force between the single micro-bulge and the interface B. Therefore, the van der Waals force decreases with increasing roughness and gradually converges towards zero.

_{o}decreases and radius of meniscus curvature r increases. The orders of magnitude of r

_{o}is much greater than r, so the orders of magnitude of 1/r

_{o}increase is much smaller than 1/r decrease. The capillary pressure Pc between the two surfaces decreases, which leads to a decrease in the capillary bridge force F

_{cap}. This change can also be reflected in Equation (30).

## 5. Double-sides Polishing Experiment Based on LSC

_{2}O

_{3}powder with 3 μm particle size. Finally, an ultra-thin sapphire wafer with a thickness of 0.17 mm was obtained. The clamping thickness of the limiter was 0.105 mm. The specific experimental parameters are shown in Table 3.

_{a}= 1.52 nm, the optimal surface roughness (R

_{a}) is 1.4 nm, 3-D surface roughness (S

_{a}) is 1.1 nm. Figure 10b shows the flatness of ultra-thin sapphire wafers based on the double-sides polishing in the LSC mothed. The flatness (PV) can reach 0.968 μm.

## 6. Conclusion

- Under the conditions of same pressure and surface spacing, the van der Waals force is mainly determined by hardness and Hamaker coefficient of material.
- The adhesion force between the solid-liquid interface is mainly depends on capillary force, and van der Waals force is almost negligible.
- The effect of capillary force is mainly affected by the volume of droplet, roughness and material. With the increasing of droplet volume, the height of completely capillary bridge formed between the two surfaces will also increasing, and the roughness and material will affect the contact angle of the surface.
- Through the LSC method, the ultra-thin sapphire wafer can obtain an average surface roughness (R
_{a}) of 1.52 nm and a flatness (PV) of 0.968 μm.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of double-sides processing by layer stacked clamping (LSC) method: (

**a**) principle of layer stacked clamping; (

**b**) schematic diagram of layer stacked double-sides processing.

**Figure 3.**Capillary bridge between curved surface and flat surface: (

**a**) the state in which the droplet is compressed between the two surfaces; (

**b**) the two surfaces are separated and the droplets form a capillary bridge.

**Figure 4.**The test platform and the measurement curve structure: (

**a**) structure diagram of triaxial force measuring platform; (

**b**) the adhesion force measurement curve obtained by the test platform.

**Figure 5.**Effect of root mean square roughness (RMS) on adhesion force: (

**a**) the droplet volume is 50 μL; (

**b**) the droplet volume is 100 μL; (

**c**) the droplet volume is 150 μL; (

**d**) relationship between rough surface area ratio and RMS.

**Figure 6.**Influence of different materials on adhesion and van der Waals force: (

**a**) comparison of experimental and theoretical values of adhesion force under different materials; (

**b**) variation of van der Waals’ theoretical value under different materials.

**Figure 7.**Layer stacked fixture and double-sides polishing equipment: (

**a**) double-sides polishing machine; (

**b**) layer stacked fixture.

**Figure 10.**Surface morphology of sapphire based on LSC after double-sides polishing: (

**a**) surface roughness measured by white light interferometer, the roughness (R

_{a}) is 1.4 nm and the 3D surface roughness (S

_{a}) is 1.1 nm; (

**b**) the flatness of ultra-thin sapphire (PV) is 0.968 μm.

Material | Sapphire | Al Alloy | Iron | 304 Stainless Steel | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||

R_{a} (nm) | 3.5 | 6.5 | 6.4 | 3.6 | 12.1 | 68.2 | 210.1 | 517.9 |

Root mean square roughness (RMS) (nm) | 4.2 | 8.2 | 7.8 | 4.4 | 14.2 | 93.0 | 255.4 | 659.0 |

Connect angle θ’ (°) | 51.2 | 72.6 | 47.1 | 60.1 | 66.7 | 82.6 | 84.8 | 86.5 |

Cutoff frequency ω_{l} | 12.5 | 12.5 | 12.5 | 12.5 | 12.5 | 4 | 1.25 | 1.25 |

High frequency ω_{h} | 400 | |||||||

Height difference H’ (μm) | - | 10 |

Parameter | Material | ||||
---|---|---|---|---|---|

Water | Sapphire | Iron | Al Alloy | 304 Stainless Steel | |

Hamaker coefficient J (10^{−20} J) | 3.7 | 15.5 | 26 | 12.6 | 21.2 |

Elastic Modulus (GPa) | - | 379 | 210 | 68.9 | 193 |

Poisson’s ratio | - | 0.309 | 0.3 | 0.33 | 0.29 |

Brinell hardness H_{b} (N/mm^{2}) | - | - | 146 | 30 | 123 |

Name | Parameter | Name | Parameter | |
---|---|---|---|---|

Sapphire | α-Al_{2}O_{3} C direction | Rotation | Upper plate (r/min) | −24 |

Diameter of sapphire (mm) | Φ50.8 | Lower plate (r/min) | 34 | |

Sapphire thickness (mm) | 0.17 | Sun gear (r/min) | 20 | |

Abrasive | SiO_{2} | Outer gear (r/min) | 0 | |

Abrasive size (nm) | 80 | pH of slurry | 11 | |

Flow rate of slurry (mL/min) | 25 | Flatness of baseplate (μm) | 0.988 | |

Quality score of slurry (%wt) | 5 | Thickness of limit tablet (mm) | 0.105 | |

Pressure (KPa/piece) | 31.6 | Time (min) | 60 |

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**MDPI and ACS Style**

Chen, Z.; Cao, L.; Yuan, J.; Lyu, B.; Hang, W.; Wang, J.
The Mechanism of Layer Stacked Clamping (LSC) for Polishing Ultra-Thin Sapphire Wafer. *Micromachines* **2020**, *11*, 759.
https://doi.org/10.3390/mi11080759

**AMA Style**

Chen Z, Cao L, Yuan J, Lyu B, Hang W, Wang J.
The Mechanism of Layer Stacked Clamping (LSC) for Polishing Ultra-Thin Sapphire Wafer. *Micromachines*. 2020; 11(8):759.
https://doi.org/10.3390/mi11080759

**Chicago/Turabian Style**

Chen, Zhixiang, Linlin Cao, Julong Yuan, Binghai Lyu, Wei Hang, and Jiahuan Wang.
2020. "The Mechanism of Layer Stacked Clamping (LSC) for Polishing Ultra-Thin Sapphire Wafer" *Micromachines* 11, no. 8: 759.
https://doi.org/10.3390/mi11080759