## 2 Finite Element Model Verification
### Analysis of Exsisting Test Data  
In order to verify the finite element analysis model, experimental data in the CMP process for several retainer rings were provided courtesy of Will Be S&T Corp. The MRR value of the wafer was obtained after the CMP process using the thin film thickness measurement system (SL-5030 manufactured by K-MAC Ltd., Daejeon, Korea). Since the authors' laboratories did not have a CMP tester and a thin film thickness measurement system, so a well-organized test for this paper could not be performed. For this reason, this test data was not used for validating the suggested final results of this paper, but for verifying the finite element analysis model. The cross section information of each retainer ring used in the test is shown in Figure 3. Different pressures were applied for each section from the multi-zone carrier head on the back of the wafer, and detailed information is shown in Table 1. The test was carried out by applying a pressure of 11.9 psi to the retainer ring and 9.3/4.4/4.5/4.7 psi in Zone 1 to 5, respectively. The test results for the six samples are shown in Figure 4 and Table 2. Using a thin film measurement system, each MRR was measured from the center of the wafer to the left and right. Thus, two results were obtained for one sample. Plus, two samples were produced and tested for the one case. Totally, four results were obtained for one sample. Since the overall MRR values are different for each sample, the average is calculated from the center of the wafer to the 120 mm distance from the center where the MRR values are relatively constant. Then, the relative displacement from the average is calculated and shown in Figure 4 and Table 2. In Table 2, in addition to the maximum relative MRR (MRMRR), the relative MRR at 7 mm distance from the wafer end (RMRR@7 mm) was also added since some semiconductor manufacturers empirically remove the edge of the wafer by 7 mm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Distance from Wafer Center [mm]**} & **Zone 1 (P1)** & **Zone 2 (P2)** & **Zone 3 (P3)** & **Zone 4 (P4)** & **Zone 5 (Ps)** \\ \cline{2-6}  & 145–150 & 128–145 & 100–128 & 40–100 & 0–40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Applied pressure distribution on wafer. Figure 2: Shape of retainer ring: (**a**) top view, (**b**) bottom view, (**c**) cross section. Figure 4: Experimental results of six samples (relative material removal rate (MRR)): (**a**) left-hand side, (**b**) right-hand side. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Unit:** \(\mathbf{\hat{A}}\) & **\#1** & **\#2** & **\#3** & **\#4** & **\#5** & **\#6** \\ \hline \multirow{3}{*}{Maximum relative MRR} & 893/182 & 1083/915 & 1097/1092 & 1148/1027 & 1215/825 & 583/585 \\  & 551/179 & 1102/954 & 158/1079 & 1133/1118 & 1554/1533 & 494/535 \\ \cline{1-1} \cline{2-6}  & [450] & [1014] & [1107] & [1110] & [1282] & [549] \\ \cline{1-1} \cline{2-6} \cline{5-6} \cline  
By comparing the MRR data of model #1, which is a basic model, and models #2, #3, and #4 with grooves added, the effect of adding a groove can be investigated. As can be seen in Table 2, the grooves generally make the MRR worsen at the wafer edge. By comparing the MRR data of model #2 and model #3, the effect of machining the round shapes can be identified. Since the deviation of the test data is not small, it is even hard to draw a conclusion, but the MRR worsens a little within the error rage. It is estimated that the contact area is reduced by the round shapes. It will be investigated further through a finite element analysis model. In addition, by comparing the MRR data of model #2 with a groove width of 8 mm and model #4 with a groove width of 12 mm, it can be confirmed that the MRR worsens due to the increase in groove size. It is estimated that as the contact area between the retainer ring and the pad decreases, the pressure applied from the retainer ring to the pad increases, resulting in a larger rebound of the pad and non-uniformity.