本研究先建立不同研磨液體積濃度及不同研磨液研磨顆粒直徑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及不同研磨顆粒直徑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研磨液體積濃度及不同研磨液研磨顆粒直徑、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，得到固定研磨液研磨顆粒直徑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。其中α與β值和固定研磨液研磨顆粒直徑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體積濃度及不同研磨液研磨顆粒直徑的二階迴歸公式k_p (x,y)，其中x為研磨液體積濃度，y為研磨顆粒直徑，這樣我們可以只用一個迴歸公式便可計算出不同體積濃度及不同研磨液研磨顆粒直徑的每分鐘研磨移除深度。最後迴歸出不同體積濃度及不同研磨顆粒直徑MRR=k_p (x,y)P^α V^β迴歸公式。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階迴歸，得到MRR=k_p P^α V^β+S_vc的迴歸公式。一般而言研磨液之研磨顆粒直徑減少會影響每分鐘研磨移除深度，本研究再利用S_pr=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，計算不同研磨顆粒直徑研磨液、不同下壓力與不同轉速的差異值S_pr，並進行S_pr的二階迴歸，得MRR=k_p P^α V^β+〖S_vc+S〗_pr的迴歸公式。本研究進行下面8個不同體積濃度，不同研磨顆粒直徑，不同下壓力及不同轉速的CMP實驗。再將室溫下不同體積濃度及不同研磨顆粒直徑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8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8個實驗的室溫下不同體積濃度及不同研磨顆粒直徑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。而固定研磨液研磨顆粒直徑在不同研磨液體積濃度下，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 )。本研究也進行MRR=k_p (x,y)P^α V^β+〖S_vc+S〗_pr的補償迴歸公式。由於如同上述由理論模擬值所迴歸出的S_vc與S_pr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_pr〗_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及不同研磨液研磨顆粒直徑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 a theoretical simulation model of abrasive removal depth of silicon wafer by chemical mechanical polishing (CMP) of pattern-free polishing pad with slurry at different volume concentrations and slurry at different abrasive particle diameters. First of all, the silicon wafer is dipped in slurry at different volume concentrations at room temperature. After that, atomic force microscopic (AFM) experiment of the silicon wafer is conducted to calculate and obtain the specific downward force energy (SDFE_reaction) values of the silicon wafer dipped in slurry at different volume concentrations. These SDFE_reaction values are substituted into the innovatively established theoretical simulation model of the abrasive removal depth per minute of silicon wafer by CMP of pattern-free polishing pad with slurry at different volume concentrations and different abrasive particle diameters. Furthermore, the paper simulates calculation of the theoretical simulation values of the abrasive removal depth per minute of silicon wafer in slurry at different volume concentrations and in slurry at different abrasive particle diameters at different downward forces and different rotational speeds.
The paper uses the obtained theoretical simulation values of the abrasive removal depth per minute of silicon wafer to perform regression analysis, obtaining the regression equation of abrasive removal depth per minute with slurry at a fixed abrasive particle diameter and different volume concentrations, MRR=k_p P^α V^β, where α and β values are the same as the α and β values of abrasive removal depth per minute with slurry at a fixed abrasive particle diameter and different volume concentrations, only that k_p is changed. Therefore, we can consider the quadratic regression equation of k_p (x,y) with slurry at different volume concentrations and with slurry at different abrasive particle diameters. For k_p (x,y), x denotes the volume concentration of slurry, and y denotes the abrasive particle diameter of slurry. In this way, only using a regression equation, we can calculate the abrasive removal depth per minute with slurry at different volume concentrations and with slurry at different abrasive particle diameter. Finally, we can regress a regression equation MRR=k_p (x,y)P^α V^β with slurry at different volume concentrations and different abrasive particle diameters.
The paper further uses the concept that 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, at different downward forces and at different rotational speeds, and the abrasive removal depth per minute obtained from the regression equation. Under the condition of a fixed downward force, the difference value S_vc performs quadratic regression, achieving a regression equation, MRR=k_p P^α V^β+S_vc. Generally speaking, reducing the abrasive particle diameter of slurry would affect the abrasive removal depth per minute. The paper further uses the concept that S_pr= Abrasive removal depth per minute obtained from theoretical simulation -(k_p P^α V^β+S_vc) to calculate the difference value S_pr with slurry at different abrasive particle diameters, at different downward forces and at different rotational speeds. In addition, the paper performs quadratic regression of S_pr, obtaining a regression equation, MRR=k_p P^α V^β+〖S_vc+S〗_pr.
The paper carries out 8 CMP experiments with slurry at different volume concentrations and different abrasive particle diameters, at different downward forces and at different rotational speeds. The theoretical simulation result of abrasive removal depth per minute in the CMP of pattern-free polishing pad with slurry at different volume concentrations and different abrasive particle diameters at room temperature is compared with the above 8 CMP experimental results of abrasive removal depth per minute. The paper calculates the individual theoretical simulation result of abrasive removal depth per minute of silicon wafer in slurry at different volume concentrations and different abrasive particle diameters at room temperature in the above 8 experiments, and further calculates the average difference ratio value, which is obtained by the paper to be around 4.2%. After deducting each of all these theoretical simulation values by the average difference ratio value of 4.2%, the paper obtains the average abrasive removal depth value per minute of the proximity experiment, and further conducts regression analysis of it with slurry at a fixed volume concentration to achieve the regression equation MRR_e=〖k_p〗_e P^(α_e ) V^(β_e ). As found in the values of 〖k_p〗_e,α_e and β_e in the obtained regression equation, when the slurry is at different volume concentrations, only the value of 〖k_p〗_e changes, but the values of α_e and β_e remain unchanged. Besides, when the slurry is at a fixed abrasive particle diameter and different volume concentrations, the obtained regression equation of average abrasive removal depth per minute of the proximity experiment is MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ). The paper also finds the compensatory regression equation MRR=k_p (x,y)P^α V^β+〖S_vc+S〗_pr. Same as the above situation, the values of S_vc and S_pr regressed from the theoretical simulation values do not have great effect on abrasive removal depth value. Therefore, achieving a compensatory regression equation of 〖S_vc〗_e and 〖S_pr〗_e from the proximity experiment’s average abrasive removal depth value per minute and MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) also does not greatly effect the results calculated from the equation. Therefore, we find that MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ) is the most convenient way for calculation of a better regression equation of the experimental average abrasive removal depth per minute of the proximity experiment. Finally, the paper carries out CMP of pattern-free polishing pad with slurry at new and different volume concentrations and with slurry at different abrasive particle diameters, which were not done in the abovementioned analytic experimental case of difference ratio value. After comparing the abrasive removal depth value per minute obtained in the new experiment, with the abrasive removal depth value per minute calculated from the regression equation MRR_e=〖k_p〗_e (x,y)P^(α_e ) V^(β_e ), the paper finds 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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