International Symposium on Semiconductor Manufacturing Intelligence (ISMI 2015) Optimizing Multiple Response Variables of CMP Process for Semiconductor Fabrication Using a Clustering Method Jaehyun Park and Dong-Hee Lee* *Corresponding author Tel.: +82-2-2220-0502; E-mail addresses: dh@hanyang.ac.kr
CMP Process Chemical and Mechanical Planarization (CMP) Process • One of unite processes for semiconductor fabrication • aims to achieve high planarity across the wafer surface (i.e., minimize roughness) Head Wafer CMP slurry Slurry Conditioner Head Polishing pad Wafer Pad Platen Schematic illustration of CMP process Definition of Roughness 응용시스템학과 2/10 Dept. of Applied Systems
Problem Statements Response Variables Process Variables • Rq: Roughness on the wafer surface • Slurry material composition (Smaller the better) (x1, x2, x3: proportions of pure slurries of which particle sizes are 30nm, 70nm, 200nm respectively) • Ra: Removal rate of material that should • Operation condition: pH of Slurry, Temperature be removed (Larger the better) • Rotating speed of wafer and platen • Rb: Removal rate of material that should • Pressure between the wafer be protected from removal (Smaller the surface and polishing pad better) Suggested CMP problem optimize 𝑆𝑏 𝐲 , 𝑆𝑐 𝐲 , 𝑆𝑟 𝐲 𝐲 s. t. 𝑦 1 + 𝑦 2 + 𝑦 3 = 100% Difficulty Three response variables should be optimized, simultaneously “Multiresponse optimization method” is needed 응용시스템학과 3/10 Dept. of Applied Systems
Three Stages for Optimization Posterior preference articulation approach to MultiResponse Surface Optimization based on Clustering method (PMRSOC) (Lee and Lee, 2015) is adopted for the optimization Stages of PMRSOC Action for the suggested CMP problem Applied Method Mixture experiments are conducted to build a 1. Model Building Simplex centroid design reliable statistical models of Rq, Ra, Rb ε -constraints method 2. Nondominated Solutions Candidate solutions are generated Generation (Haimes et al., 1971) Best solution is selected from the candidate 3. Best Solution Selection K-means clustering solutions 응용시스템학과 4/10 Dept. of Applied Systems
Stage 1. Model Building Simplex of three input variables and its coordinate system Design points 𝑺 𝒓 𝑺 𝒃 𝑺 𝒄 𝑧 𝑆𝑏 𝐲 = 916.9𝑦 1 + 731.1𝑦 2 + 325.0𝑦 3 + 42.6𝑦 1 𝑦 2 + ( 𝒚 𝟐 , 𝒚 𝟑 , 𝒚 𝟒 ) 2107.1𝑦 1 𝑦 3 + 664.5𝑦 2 𝑦 3 + 10473.0𝑦 1 𝑦 2 𝑦 3 , (1, 0, 0) 909.21 122.86 8.57 (0, 1, 0) 728.95 102.52 10.20 𝑧 𝑆𝑐 𝐲 = 125.56𝑦 1 + 101.16𝑦 2 + 31.61𝑦 3 + 16.66𝑦 1 𝑦 2 + (0, 0, 1) 323.90 32.33 13.26 335.36𝑦 1 𝑦 3 + 119.56𝑦 2 𝑦 3 + 1327.48𝑦 1 𝑦 2 𝑦 3 , (1/2, 1/2, 0) 824.82 116.18 9.20 𝑧 𝑆𝑟 𝐲 = 8.6𝑦 1 + 10.24𝑦 2 + 13.27𝑦 3 − 0.6𝑦 1 𝑦 2 − (1/2, 0, 1/2) 1138.98 160.44 8.49 (0, 1/2, 1/2) 690.93 98.35 7.72 9.64𝑦 1 𝑦 3 − 15.97𝑦 2 𝑦 3 − 29.84𝑦 1 𝑦 2 𝑦 3 . (1/3, 1/3, 1/3) 1325.55 185.77 6.47 (4/6, 1/6, 1/6) 1272.40 181.59 7.68 The R 2 ( R 2 adj ) of the three models for Ra , Rb , and Rq are 99.64% (1/6, 4/6, 1/6) 1042.86 139.24 8.00 (98.91%), 99.38% (98.13%), and 99.59% (98.78%), respectively, (1/6, 1/6, 4/6) 1008.46 132.95 8.65 응용시스템학과 5/10 Dept. of Applied Systems
Stage 2. Nondominated Solutions Generation Definition of nondominated solution • A solution 𝐲 is said to be nondominated if and only if there does not exist any other 𝐲 ∈ 𝛁 such that 𝑧 𝑆𝑏 ( x ) ≥ 𝑧 𝑆𝑏 ( 𝐲 ), 𝑧 𝑆𝑐 ( x ) ≤ 𝑧 𝑆𝑐 ( 𝐲 ), 𝑧 𝑆𝑟 ( x ) ≤ 𝑧 𝑆𝑟 ( 𝐲 ), and x ≠ 𝐲 . ε -constraints method: Maximize 𝑧 𝑆𝑏 (𝐲) by restricting 𝑧 𝑆𝑐 (𝐲) and 𝑧 𝑆𝑟 (𝐲) Maximize 𝑧 𝑆𝑏 𝐲 s. t. 𝑧 𝑆𝑐 𝑦 ≤ 𝜁 𝑆𝑐 , 𝑧 𝑆𝑟 𝑦 ≤ 𝜁 𝑆𝑟 14 12 𝑧 𝑆𝑟 10 8 150 100 𝑧 𝑆𝑐 0 50 400 800 0 𝑧 𝑆𝑏 1200 200 nondominated solutions 응용시스템학과 6/10 Dept. of Applied Systems
Stage 3. Best Solution Selection (1/3) Grouping and Selection Strategy Start Step 1: Divide the nondominated solutions into two groups by using K-means clustering Step 2: Select a preferred group Step 2: Are the solutions no of the selected group close to each other? yes Step 3: Select a final solution from the solutions of the selected group End 응용시스템학과 7/10 Dept. of Applied Systems
Stage 3. Best Solution Selection (2/3) K-means Clustering • attempts to group a set of nondominated solutions in such a way that the solutions in the same group are similar to each other than to those in other groups. • Algorithm 1: Determine K value 2: Select K solution randomly for the set of nondominated solutions as the initial centroids 3: repeat 4: Form K clusters by assigning all solutions to the closest centroid 5: Update the centroid of each cluster 6: until the centroids don’t change 응용시스템학과 8/10 Dept. of Applied Systems
Stage 3. Best Solution Selection (3/3) Summary of iterations of Steps 1 and 2 Group 1 Group 2 Ranges of Selected Iteration solutions of group Centroid Number of Centroid Number of selected group ( ŷ Ra , ŷ Rb , ŷ Rq ) ( ŷ Ra , ŷ Rb , ŷ Rq ) Solutions Solutions 1 (911.79, 125.15, 8.12) 82 (533.36, 70.62, 9.21) 118 1 (521.56, 70.74, 3.24) 2 (820.03, 112.70, 8.45) 52 (1070.85, 146.72, 7.53) 30 1 (247.69, 30.74, 2.72) 3 (768.41, 106.47, 8.57) 32 (902.63, 122.67, 8.26) 20 1 (114.24, 11.54, 2.52) 4 (797.51, 110.86, 8.48) 16 (739.32, 102.07, 8.67) 16 2 (35.34, 3.67, 2.39) 5 (742.14, 102.53, 8.96) 12 (730.84, 100.71, 7.80) 4 1 (29.11, 2.93, 2.16) 6 (737.09, 102.94, 9.30) 8 (752.24, 101.7, 8.27) 4 2 (12.93, 2.80, 0.72) Final 4 solutions in Step 3 Performance at the existing slurry Input variable Response Input variable Estimated Response ( 𝑺 𝒃 , 𝑺 𝒄 , 𝑺 𝒓 ) (x1, x2, x3) (x1, x2, x3) ( ŷ Ra , ŷ Rb , ŷ Rq ) (0%, 100%, 0%) (728.95, 102.52, 10.20) (3.24%, 34.97%, 61.80%) (745.72, 101,00, 8.20) (4.00%, 31.33%, 64.67%) (750.29, 101.00, 8.40) (4.75%, 28.10%, 67.15%) (754.30, 101.00, 8.60) (0.20%, 63.22%, 36.57%) (758.65, 103.80, 7.88) 응용시스템학과 9/10 Dept. of Applied Systems
Concluding Remarks Summary • The optimal blend of the mixture slurry was determined by applying PMRSOC • We showed that 𝑆𝑏 , 𝑆𝑐 , and 𝑆𝑟 at the obtained setting are better than those at the existing pure slurry composition. Limitation • No confirmation run Future research issues • Effects of other process variables such as pH of Surry, Temperature need to be investigated • Variability of Ra, Rb, and Rq needs to be investigated 응용시스템학과 10/10 Dept. of Applied Systems
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