本研究先建立室溫下不同體積濃度之無花紋研磨墊化學機械拋光矽晶圓的研磨移除深度理論模擬模式。我們將矽晶圓浸泡在常溫下不同體積濃度研磨液後,接著進行原子力顯微鏡實驗,計算得出浸泡室溫不同體積濃度研磨液的矽晶圓比下壓能值,再將這些比下壓能值代入創新建立的不同體積濃度之無花紋研磨墊化學機械拋光矽晶圓的每分鐘研磨移除深度理論模擬模式。進而模擬計算出室溫下不同研磨液體積濃度、不同下壓力及不同轉速所得矽晶圓每分鐘研磨移除深度之理論模擬值。
本研究再利用所得之矽晶圓每分鐘研磨移除深度之理論模擬值進行迴歸分析，得到固定體積濃度之每分鐘研磨移除深度的迴歸公式MRR=k_p P^α V^β公式後，我們發現在不同研磨液體積濃度下只有k_p改變，α與β值固定不變。所以我們進一步用一個考慮不同研磨液體積濃度的二次迴歸公式k_p (x)，其中x為研磨液體積濃度，代替固定研磨液體積濃度的迴歸公式，這樣我們可以只用一個迴歸公式便可計算出不同研磨液體積濃度的每分鐘研磨移除深度。最後迴歸出不同體積濃度MRR=k_p (x)P^α V^β迴歸公式。
本研究用研磨移除深度的理論模式模擬計算出研磨液體積濃度20%、30%、40%、50%，不同下壓力1psi、1.5psi、2psi、2.5psi、3psi與不同轉速20rpm、30rpm、40rpm、50rpm、60rpm的每分鐘研磨移除深度的理論模擬之MRR值。本研究再利用S_vc=理論模擬得出的每分鐘研磨移除深度-k_p P^α V^β的觀念，計算出固定研磨液體積濃度、不同下壓力與不同轉速的每分鐘研磨移除深度的理論模擬值與由迴歸公式得出之每分鐘研磨移除深度的差異值S_vc。我們將差異值S_vc以固定下壓力的條件下進行二次線性迴歸，得到MRR=k_p P^α V^β+S_vc的迴歸公式。並且比較後發現理論模擬所得的研磨移除深度和由MRR=k_p (x)P^α V^β迴歸公式得出之MRR值其差異值在加上S_vc補償迴歸值後影響效果很小，所以使用上我們可以忽略S_vc補償公式，以MRR=k_p (x)P^α V^β公式即可求得所需之每分鐘研磨移除深度。
本研究先進行下面6個不同體積濃度，不同下壓力及不同轉速的CMP實驗 (1) 20%、3psi，60rpm ; (2) 30%、3psi，60rpm ; (3) 50%、3psi，60rpm ; (4) 40%、2psi、40rpm ; (5) 50%、2psi、40rpm及(6) 50%、1psi、60rpm，再將室溫下不同體積濃度研磨液之無花紋研磨墊化學機械拋光的每分鐘研磨移除深度之理論模擬結果與上述6個化學機械拋光的實驗結果之每分鐘研磨移除深度做比較。計算出上述6個實驗的室溫下不同體積濃度研磨液拋光矽晶圓的個別每分鐘研磨移除深度理論模擬值與實驗之平均每分鐘研磨移除深度值，再計算其平均差異比例值，本文求出平均差異比例值約為4.2%。本文將所有理論模擬值在減去平均差異比例值4.2%後得到接近實驗之平均每分鐘研磨移除深度值，進一步將其迴歸分析固定體積濃度的MRR_e=〖k_p〗_e P^(α_e ) V^(β_e )迴歸公式。由迴歸所得之公式的〖k_p〗_e、α_e 和β_e 之值，可發現在不同研磨液體積濃度下，只有〖k_p〗_e改變，α_e 與β_e 值固定不變。所以我們進一步迴歸分析不同體積濃度x的MRR_e=〖k_p〗_e (x)P^(α_e ) V^(β_e )迴歸公式。由於如同上述由理論模擬值所迴歸出的S_vc值影響研磨移除深度值不大，故由〖S_vc〗_e補償迴歸公式對於公式的計算結果影響也不大。所以我們發現MRR_e=〖k_p〗_e (x)P^(α_e ) V^(β_e )為最方便計算每分鐘研磨移除深度實驗值的較佳迴歸公式。最後本研究另外進行不在前面做差異比例值分析實驗案例中的新的不同體積濃度研磨液的無花紋研磨墊化學機械拋光實驗，將新的實驗所得每分鐘研磨移除深度值與迴歸公式MRR_e=〖k_p〗_e (x)P^(α_e ) V^(β_e )計算所得之每分鐘研磨移除深度值進行比較後，發現其差異很小，由此可驗證迴歸公式MRR_e=〖k_p〗_e (x)P^(α_e ) V^(β_e )為合理且實用。

In the beginning the paper establishes a theoretical simulation model of abrasive removal depth of silicon wafer dipped in slurry at different volume concentrations at room temperature after chemical mechanical polishing (CMP) of pattern-free polishing pad. After we have dipped silicon wafer in slurry at different volume concentrations at room temperature, we make atomic force microscopy experiment, and calculate the specific down force energy (SDFE) values of silicon wafer dipped in slurry at different volume concentrations at room temperature. Then the paper substitutes these SDFE values in an innovatively established theoretical simulation model of abrasive removal depth of silicon wafer dipped in slurry at different volume concentrations after chemical mechanical polishing (CMP) of pattern-free polishing pad. Furthermore, through simulation, the paper calculates the theoretical simulation values of the obtained abrasive removal depths per minute of silicon wafer dipped in slurry at different volume concentrations at room temperature, under different downward forces and at different rotational speeds.
The paper subsequently uses the obtained theoretical simulation values of the abrasive removal depths per minute of silicon wafer to perform regression analysis. After achieving a regression equation of abrasive removal depth per minute with slurry at a fixed volume concentration: MRR = kpPV, we find that when slurry is at different volume concentrations, only kp is changed, but with  and  values remaining unchanged. Therefore, we further use a secondary regression equation kp(x) that considers different volume concentrations of slurry, with x being the volume concentrations of slurry. This secondary regression equation replaces the regression equation with a fixed volume concentration of slurry. In this way, using only one regression equation, we can calculate the abrasive removal depth per minute with slurry at different volume concentrations. Finally, a regression equation with slurry at different volume concentrations can be obtained: MRR = kp (x)PV.
The paper uses a theoretical model of abrasive removal depth for simulation, and calculates the abrasive removal depth per minute value obtained by theoretical simulation when slurry is at volume concentrations of 20%, 30%, 40% and 50%, under different downward forces of 1psi, 1.5psi, 2psi, 2.5psi and 3psi, and at different rotational speeds of 20rpm, 30rpm, 40rpm, 50rpm and 60rpm. After that, using the concept that Svc = the abrasive removal depth per minute obtained by theoretical simulation  kpPV, the paper calculates the difference value Svc between the theoretical simulation values with slurry at a fixed volume concentrations, under different downward forces and at different rotational speeds, and abrasive removal depth per minute obtained from the regression equation.Using the difference value Svc under the condition of a fixed downward force, we perform secondary linear regression, and then achieves a regression equation: MRR = kpPV + Svc . After comparison is made, we find that when the difference value between the abrasive removal depth per minute obtained from the theoretical simulation and the MRR value obtained from the regression equation MRR = kp (x)PV, adds the compensatory regression value Svc , the effect of influence is very small. Therefore, in times of use, we can ignore the compensatory equation with Svc . Simply using the equation MRR = kp (x)PV, we can just find the required abrasive removal depth per minute.
The paper firstly conducts CMP experiments using the following 6 different volume concentrations of slurry, different downward forces and different rotational speeds: (1) 20%, 3psi, 60rpm; (2) 30%, 3psi, 60rpm; (3) 50%, 3psi, 60rpm; (4) 40%, 2psi, 40rpm; (5) 50%, 2psi, 40rpm; and (6) 50%, 1psi, 60rpm. Then the paper compares the theoretical simulation results of abrasive removal depths per minute with slurry at different volume concentrations after CMP of pattern-free polishing pad, with the results of abrasive removal depths per minute obtained from the above 6 CMP experiments. Through calculation, the paper obtains the average abrasive removal depth value per minute between the theoretical simulation value and experimental value of individual abrasive removal depth of silicon wafer per minute with slurry at different volume concentrations at room temperature as mentioned in the above 6 experiments, and then calculates the average difference ratio value. The average difference ratio value calculated by the paper is around 4.2%. After the paper deducts each of the theoretical simulation values by the average difference ratio value of 4.2%, an average abrasive removal depth value per minute being close to the experimental result is obtained. Furthermore, the paper use these close to experimental result obtained to make regression analysis for slurry at a fixed volume concentration, and achieves a regression equation: MRR = kpePeVe. From the values of kpe, _e and _e in the equation obtained from regression, we can find that when slurry is at different volume concentrations, only kpe is changed, but with _e and _e values remaining fixed and unchanged. Therefore, we further makes regression analysis for slurry at different volume concentrations x, and achieves a regression equation: MRRe = kpe(x)PeVe. Since it is understood from the above that the Svc value regressed from the theoretical simulation value does not influence the abrasive removal depth value per minute much, the compensatory regression equation with Svce also will not greatly influence the calculation result of the equation. Therefore, we find that MRRe = kpe(x)PeVe is the most convenient and a better regression equation for calculation of the experimental value of abrasive removal depth per minute. Finally, the paper conducts new CMP experiments of pattern-free polishing pad with slurry at different volume concentrations, referring to the experimental cases without making analysis of difference ratio value as done above. After the abrasive removal depth values per minute obtained from the new experiments are compared with the abrasive removal depth values per minute obtained from calculation by the regression equation MRRe = kpe(x)PeVe, the paper finds that the difference in between is very small. As seen from here, it is proved that the regression equation MRRe = kpe (x)PeVe is feasible and practical.

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