本研究先建立不同研磨液溫度及不同研磨液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。我們先將矽晶圓浸泡在不同溫度及不同體積濃度研磨液後,接著進行原子力顯微鏡實驗,計算得出浸泡不同溫度及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溫度及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。進而模擬計算出不同溫度及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，得到固定研磨液溫度與不同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。而固定研磨液體積濃度在不同研磨液溫度下，所得之每分鐘研磨移除深度的迴歸公式MRR=k_p P^α V^β公式，其中α與β值和固定研磨液溫度與不同體積濃度之每分鐘研磨移除深度的α與β值相同，只有k_p改變。所以我們考慮不同研磨液溫度及不同研磨液體積濃度的二次迴歸公式k_p (x,y)，其中x為研磨液體溫度，y為研磨液體積濃度，這樣我們可以只用一個迴歸公式便可計算出不同溫度及不同研磨液體積濃度的每分鐘研磨移除深度。最後迴歸出不同溫度及不同體積濃度MRR=k_p (x,y)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溫度23℃、30℃、40℃、50℃，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,y)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,y)P^α V^β公式即可求得所需之每分鐘研磨移除深度。一般而言研磨液之溫度提升會影響每分鐘研磨移除深度，本研究再利用S_tm=理論模擬得出的每分鐘研磨移除深度-(k_p P^α V^β+S_vc)的觀念，的方式計算不同溫度研磨液、不同下壓力與不同轉速的差異值S_tm。
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。而固定研磨液體積濃度在不同研磨液溫度下，所得之接近實驗之平均每分鐘研磨移除深度迴歸公式MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e )公式，其α_e 和β_e 值和固定研磨液溫度不同研磨液體積濃度的α_e 與β_e值相同，只有〖k_p〗_e改變。所以我們進一步迴歸分析不同研磨液溫度x及不同研磨液體積濃度y的MRR_e=〖k_p〗_e (x,y)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與S_tm值影響研磨移除深度值不大，故由接近實驗之平均每分鐘研磨移除深度值MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e )得到的〖S_vc〗_e與〖S_tm〗_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,y)P^(α_e ) V^(β_e )為最方便計算接近實驗之平均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,y)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,y)P^(α_e ) V^(β_e )為合理且實用。

The paper firstly establishes the theoretical simulation model of abrasive removal depth of silicon wafer by chemical mechanical polishing (CMP) using pattern-free polishing pad at different slurry temperatures and different slurry volume concentrations. First of all, we dip silicon wafer in slurry at different temperatures and different volume concentrations, and then perform atomic force microscopic (AFM) experiment to calculate the specific down force energy (SDFE) values of silicon wafer dipped in slurry at different temperatures and different volume concentrations. These SDFE values are substituted in an innovatively established theoretical simulation model of abrasive removal depth of silicon wafer per minute by CMP using pattern-free polishing pad at different temperatures and different volume concentrations. Furthermore, we can make simulative calculation of the theoretical simulation values of abrasive removal depth of silicon wafer per minute obtained at different slurry temperatures, different slurry volume concentrations, different down forces and different rotational speeds.
The paper uses the obtained theoretical simulation values of abrasive removal depth of silicon wafer per minute to perform regression analysis, and achieves a regression equation MRR=k_p P^α V^β of abrasive removal depth of silicon wafer per minute at a fixed slurry temperature and different slurry volume concentrations. After that, we find that under different slurry volume concentrations, only k_p changes, but α and β values are fixed and unchanged. And when slurry is at a fixed volume concentration and different temperatures, in the obtained regression equation MRR=k_p P^α V^β of abrasive removal depth of silicon wafer per minute, α and β values are the same as the α and β values of abrasive removal depth per minute at a fixed slurry temperature and different slurry volume concentrations, and only k_p is changed. Therefore, we consider a quadratic regression equation k_p (x,y) at different slurry temperatures and different slurry volume concentrations. In this equation, x denotes the slurry temperature and y denotes the slurry volume concentration. In this way, using a regression equation only, we can calculate the abrasive removal depth per minute at different slurry temperatures and different slurry volume concentrations. Finally, we can regress a regression equation MRR=k_p (x,y)P^α V^β with slurry at different temperatures and different volume concentrations.
The paper uses a theoretical model of abrasive removal depth to make simulative calculation of the theoretical simulated MRR values of abrasive removal depth per minute at different slurry temperatures 23oC, 30oC, 40oC, 50oC, different slurry volume concentrations 20%, 30%, 40%, 50%, different down forces 1psi, 1.5psi, 2psi, 2.5psi, 3psi, and different rotational speeds 20rpm, 30rpm, 40rpm, 50rpm, 60rpm. Then the paper uses the concept, S_vc=Abrasive removal depth per minute obtained from theoretical simulation-k_p P^α V^β, to calculate the difference value S_vc between the theoretical simulation value of abrasive removal depth per minute with slurry at a fixed volume concentration, different down forces and different rotational speeds, and the abrasive removal depth per minute obtained from the regression equation.
We make the difference value S_vc perform quadratic linear regression under the condition of a fixed down force, achieving a regression equation MRR=k_p P^α V^β+S_vc. After comparison, we find that when the difference value between the abrasive removal depth per minute obtained from theoretical simulation and the MRR value obtained from the regression equation MRR=k_p (x,y) P^α V^β is added with the compensated regression value of S_vc, the influential impact is small. Therefore, in terms of use, we can neglect the compensation equation of S_vc, but use the equation MRR=k_p (x,y) P^α V^β only to obtain the required abrasive removal depth per minute. Generally speaking, temperature rise of slurry would affect the abrasive removal depth per minute. The paper further uses the concept, S_tm=Abrasive removal depth per minute obtained from theoretical
simulation-(k_p P^α V^β+S_vc), to calculate the difference value S_tm at different slurry temperatures, different down forces and different rotational speeds. The paper firstly performs CMP experiments with six combinations of different volume concentrations, different down 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 theoretical simulation results of abrasive removal depth per minute of silicon wafer by CMP using pattern-free polishing pad at room temperature and different volume concentrations are compared with the above six CMP experimental results of abrasive removal depth per minute. The paper calculates the theoretical simulation value of individual abrasive removal depth per minute of silicon wafer at room temperature with slurry at different volume concentrations in the above six experiments, and the experimental value of average abrasive removal depth per minute, and further calculates the average difference ratio, which is calculated by the paper to be 4.2%.
The paper lets all the theoretical simulation values subtract the average difference ratio 4.2%, achieving an average abrasive removal depth value per minute that is close to the experimental value. Furthermore, regression analysis is made for the regression equation of the average abrasive removal depth per minute that is close to the experiment value MRR_e=〖k_p〗_e P^(α_e ) V^(β_e ) at a fixed volume concentration. From the values of 〖k_p〗_e, α_e and β_e in the equation obtained from regression, we can find that under different slurry volume concentrations, only 〖k_p〗_e is changed, and the α_e and β_e values are fixed and unchanged. And under a fixed slurry volume concentration and different slurry temperatures, in the obtained regression equation MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) of average abrasive removal depth per minute that is close to the experimental value, the α_e and β_e values are the same as the α_e and β_e values under a fixed slurry temperature and different slurry volume concentrations, and only 〖k_p〗_e is changed. Therefore, we further make regression analysis of the regression equation MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) under different slurry temperatures x and different slurry volume concentrations y. As known from the above, the S_vc and S_tm values regressed from the theoretical simulation values do not greatly affect the abrasive removal depth value. Therefore, the compensated regression equation of 〖S_vc〗_e and 〖S_tm〗_e obtained from MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) of the average abrasive removal depth value per minute being close to the experimental value does not greatly affect the calculation result of the equation. Thus, we find that MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) is the most convenient and a better regression equation to calculate the experimental average abrasive removal depth value per minute being close to the experimental value. Finally, the paper performs another new CMP experiment with pattern-free polishing pad at different volume concentrations, without making difference ratio analysis in advance in the experimental case. After comparing the abrasive removal depth value per minute obtained from the new experiment with the abrasive removal depth value per minute obtained from calculation of the regression equation MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ), we find that the difference in between is very small. As proved from here, the regression equation MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) is reasonable and practical.

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