# Material Removal Model for Lapping Process Based on Spiral Groove Density
## 5 Conclusions  
The oil film thickness and frictional force during the lapping process were expressed in terms of the inverse number of the groove density, which showed a similar tendency as the oil film thickness and frictional force on the Stribeck curve. This indicates that, under the experimental conditions in this study, the wafer and platen are separated by the oil film when the groove density is lower than 0.3, resulting in a mixed lubrication condition in which only some of the large abrasives participate in lapping. At a groove density of 0.3 or higher, a solid-solid contact is made with no oil film present, resulting in a boundary lubrication condition where the wafer and the abrasives on the platen are in direct contact. The CGD, at which the lapping characteristics change with oil film thickness, was found to be 0.3 under the experimental conditions in this study. CGD is affected by the average abrasive size, distribution, pressure, and velocity.  
Figure 15: Comparison of modified theoretical material removal rate with experimental results according to groove density.  
The results of the material removal model based only on indentation theory and the experimental results were consistent at CGD or higher but showed a significant difference below CGD because the model failed to reflect the lubrication condition at the contact interface. To reflect the participation of abrasives depending on the oil film thickness in the contact model, the solid-solid contact model used in the CMP process was adopted, and pad asperities were substituted with the abrasives present on the platen. Based on the abrasive particles' size measurement, normal distribution and cumulative distribution functions were derived, and the probability function of abrasive participation as a function of the oil film thickness was derived based on the oil film function. The number of abrasives varying with oil film thickness was expressed by multiplying the derived probability function of abrasive participation by the number of active abrasives from the previous model, which confirmed that the rapid change in the material removal rate below CGD was being reflected.  
The results indicate that changes in groove density affect the lubrication condition at the contact interface as well as the probability of abrasive participation by varying the oil film thickness. When fabricating the groove for a lapping platen, a groove density at CGD or higher should be selected. As a parameter related to the oil film thickness and abrasive size, CGD can vary with the Sommerfeld number, abrasive size, and abrasive distribution. The higher the groove density, the easier it is to avoid CGD, and the higher is the material removal rate. However, the ratio of the platen land in direct contact with the wafer reduces, increasing the wear and decreasing the lifespan of the platen. Therefore, the economic benefit with respect to the higher material removal rate and shorter life span of the platen should be weighted when designing the groove density.  
Conceptualization, T.L. and H.K.; methodology, T.L., H.C., and S.L.; validation, D.K. and S.L.; investigation, H.C.; resources, D.K.; writing--original draft preparation, T.L.; writing--review and editing, H.J. and H.C.; supervision, H.K.; project administration, H.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Trade, Industry, and Energy of the Republic of Korea (grant number 20013589).  
Not applicable.  
The authors declare no conflict of interest.