# Material Removal Model for Lapping Process Based on Spiral Groove Density
## 2 Materials and Methods
### Experimental Equipment and Conditions  
A lapping experiment was conducted with varying groove densities to understand the effect of groove density on the material removal rate during the lapping process. A rectangular groove with a spiral pattern was selected, and the depth and width of the grooves were fixed at 1 and 2 mm, respectively, while the pitch was set as a variable to vary the groove density from 0.1 to 0.7. The lapping equipment shown in Figure 6 was used for the lapping experiment; it consists of a lapping head to attach a single 100 mm wafer, an air cylinder to pressurize the lapping head, a 300 mm diameter platen, a slurry nozzle to supply the slurry, and a dresser to distribute the slurry. Polycrystalline diamond particles with an average size of 3.1 \(\upmu\)m were used as the abrasive, and the platen was prepared in copper resin by sintering a mixture of copper powder and resin. The experimental conditions, such as the pressure applied to the wafer, rotational speed of the platen, and flow rate of the slurry, were fixed so that only the effect of changes in groove density was considered. The surface of the platen was refined through a facing process to make the surface roughness and platen shape consistent. Prior to the experiment, a stabilization process was conducted for 10 min under the same conditions as the lapping experiment. A contact-type displacement sensor (GT2-A12 KL, Keyence, Osaka, Japan) with a resolution of 0.1 \(\upmu\)m was used to measure the thickness of the wafer. The average thickness was obtained by measuring the wafer thickness at 21 locations before and after lapping, and the material removal rate was calculated by dividing the change in the average thickness by the processing time. Detailed experimental conditions are listed in Table 1.  
Figure 5: Theoretical material removal rate as a function of groove density. Figure 6: Lapping equipment. In the lapping process, it is believed that the changes in groove density affect the material removal rate because the contact interface condition between the wafer and platen changes. The contact interface condition is related to the hydroplaning phenomenon caused by the slurry. [PERSON] [24] used a laser-induced fluorescence device to observe changes in oil film thickness based on the process conditions, and [PERSON] [25] used ultrasonic waves to measure and analyze the oil film thickness during the process. In this study, to analyze the changes in the contact interface condition with groove density, the oil film thickness was measured by constructing the system shown in Figure 7a and using an inductive displacement sensor (Millimar 1318, Mahr, Goettingen, Germany) with a 0.1 \(\upmu\)m resolution. For the measurement of oil film thickness, the displacement of the lapping head was first measured for 10 s under dry conditions without supplying the slurry. To observe changes in the thickness of the oil film, the displacement of the lapping head was measured for 30 s under wet conditions while supplying the slurry to the platen until the development of the oil film stabilized under the same conditions as the lapping experiment. The oil film thickness was calculated using the difference in the displacement of the head measured under dry and wet conditions. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Condition** \\ \hline Wafer & Sapphire, \(\Phi\)100 mm \\ \hline Lapping plate & Copper resin, \(\Phi\)300 mm \\ \hline Slurry & Polycrystalline diamond, average size 3.1 \(\upmu\)m \\ \hline Flow rate & 4 mL/min \\ \hline Pressure & 400 gf/cm\({}^{2}\) \\ \hline Velocity & 80 rpm \\ \hline Depth & 1.0 mm \\ \hline Width & 2.0 mm \\ \hline Pitch [mm] & 20.0 & 15.0 & 10.0 & 6.7 & 5.0 & 4.0 & 3.3 & 2.9 \\ \hline Density (\(d_{g}\)) & 0.10 & 0.13 & 0.20 & 0.30 & 0.40 & 0.50 & 0.61 & 0.69 \\ \hline \end{tabular}
\end{table} 
Table 1: Experimental conditions. Figure 7: Measuring method: (**a**) oil film thickness and (**b**) frictional force.