# Material Removal Model for Lapping Process Based on Spiral Groove Density
## 4 Modified Material Removal Model
### Material Removal Model Considering Oil Film Thickness  
In the lapping process, the solid-solid contact occurs between the wafer and abrasives, which are present between the platen and wafer. In the material removal model derived from indentation theory, the number of abrasives in contact with the wafer was assumed to be proportional to the contact area between the platen and wafer, but changes in the oil film thickness at the contact interface were not considered. To take these changes into account, it was assumed that the size of the abrasives present on the platen has a normal distribution and the number of abrasives in contact would vary with the oil film thickness. The size distribution of abrasives can be expressed as shown in Equation (10):  
\[f(r_{d})=\frac{1}{\sqrt{2\pi}\cdot\sigma}e^{-0.5\left(\frac{d-m}{\sigma} \right)^{2}}, \tag{10}\]  
where \(r_{d}\) [\(\upmu\)m], \(m\) [\(\upmu\)m], and \(\sigma\) are the abrasive size, average abrasive size, and standard deviation of the abrasive size, respectively. The cumulative distribution of the abrasive size is an integral of Equation (10), as expressed in Equation (11):  
\[g(r_{d})=\int_{0}^{r_{d},\ max}f(r_{d})dr_{d}. \tag{11}\]  
The average abrasive size used in the experiment was 3.13 \(\upmu\)m, the standard deviation was 0.55, the minimum size was 1.06 \(\upmu\)m, and the maximum size was 5.45 \(\upmu\)m. Figure 13 shows the normal distribution and cumulative distribution of the abrasive size calculated using Equations (10) and (11). Figure 13: (**a**) Normal distribution of abrasive size; (**b**) cumulative distribution of abrasive size. Figure 14a shows the distribution relationship between the oil film thickness and the abrasive size in order to define the probability of abrasive participation as a function of the film thickness. Since the film thickness is 6.5 \(\upmu\)m and the abrasive size has a distribution of 1.06-5.45 \(\upmu\)m at a groove density of 0.2, no abrasives participate in lapping and all of the load is supported by the hydrodynamic pressure. As the groove density increases from 0.2 to 0.3, the film thickness increases from 6.5 to 1.3 \(\upmu\)m, and the number of abrasives participating in polishing drastically increases. In other words, when a line is drawn at the film thickness parallel to the \(x\)-axis, the area surrounded by the parallel line and the abrasive distribution curve represents the probability of abrasive participation. Therefore, the probability of abrasive participation, \(k_{f}\), as a function of the film thickness can be expressed as shown in Equation (12):  
\[k_{f}\Big{(}t_{f}\Big{)}=1-\int_{0}^{t_{f}}\frac{1}{\sqrt{2\uppi}\cdot\sigma} e^{-0.5\left(\frac{t_{f}-m}{\sigma}\right)^{2}}\text{dt}_{f}. \tag{12}\]  
Here, the film thickness, \(t_{f}\left[\upmu\right]\), is expressed in terms of the groove density, \(d_{g}\), as shown in Equation (13) and is derived from the experimental result. \[t_{f}\big{(}d_{g}\big{)}=32.7e^{-9.6d_{g}}. \tag{13}\]  
Figure 14b shows the probability of abrasive participation with film thickness calculated using Equation (12). To calculate the number of active abrasives as a function of the film thickness, the probability of abrasive participation with film thickness was calculated based on Figure 14b and Equation (12), which was multiplied by the number of active abrasives from the previous material removal model. In the previous material removal model, the material removal rate of a single abrasive was first obtained, and the number of active abrasives in the entire wafer area was calculated to determine the overall material removal rate. In the modified material removal model, the probability of abrasive participation, \(k_{f}\), is introduced to take into account the changes in the film thickness.