統計在半導體產業的應用 -- Basic Statistic Methods
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Transcript of 統計在半導體產業的應用 -- Basic Statistic Methods
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半導體產業常用的統計方法
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大綱
常用的統計方法 各產業的挑戰 進階分析用的方法
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基本統計量
Describe statistics
-- mean/median/std/Q1/Q3/percentile Display
-- box/histogram/CDF/P-P/Q-Q
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統計方法
假設檢定 -- 要檢定什麼 ? Null=? Alternative=?
迴歸模型 -- 自變數 ? 應變數 ? 連續 / 非連續
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Design house
資料來源 :
--WAT( 代工廠 )
--CP and final test( 封測廠 ) 工程師的問題 :
-- 那些因素影響到 final yield 統計上的問法 ?
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可用的統計方法
Correlation coefficient Regression Testing hypothesis
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統計上的問題
那些變數與 final yield 有關
LotID Waf CP1 Yield CP1 Bin1 CP1 Bin2 CP1 Bin3 CP1 Bin4GU3517.1 1 77.67 77.41 5.65 8.31 0.66GU3517.1 2 88.33 88.04 1.33 4.32 0.33GU3517.1 3 81 80.73 3.99 7.64 1GU3517.1 4 78.33 78.07 5.98 3.65 0GU3517.1 5 74.67 74.42 7.97 5.65 1.33GU3517.1 6 83.67 83.39 4.98 4.98 1GU3517.1 7 82 81.73 7.64 4.32 1GU3517.1 8 82.33 82.06 4.32 6.31 0.66GU3517.1 9 78.67 78.41 3.99 7.31 1.33GU3517.1 10 76 75.75 5.65 6.98 0.66GU3517.1 11 77 76.74 5.98 8.64 0.66GU3517.1 13 78 77.74 12.62 4.32 0.33GU3517.1 14 85.33 85.05 2.33 4.65 0.33GU3517.1 15 79.67 79.4 5.32 6.98 0.66GU3517.1 16 78.33 78.07 10.63 4.65 0.33GU3517.1 17 84 83.72 3.65 4.32 0.66GU3517.1 18 85 84.72 3.32 4.98 0.33GU3517.1 19 71.67 71.43 13.95 7.97 0
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Correlation CoefficientParameter rBV_N43_V_9..35 NABV_N4_V_9..117 NABV_P43_V_9..3 0.047507BV_P4_V_9..117 NACONTI_M1_OHM.144 0.233358CONTI_M2_OHM.18 0.219298CONTI_M3_OHM.18 0.137096CONTI_M4_OHM.18 0.065599CONTI_M5_OHM.18 0.227058CONTI_M6_OHM.18 0.313128CONTI_PO_OHM.117 0.128133Isat_N43_mA_9..35 0.027397Isat_N4_mA_9..117 0.015288Isat_P43_mA_9..3 0.234876Isat_P4_mA_9..117 -0.03807MIM_CAP_fF.um2 0.020579Rc_N.PO_OHM.SE.144 0.053797Rc_N._OHM.SE.144 -0.05793
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RegerssionEstimate Std. Error t value Pr(>|t|)
(Intercept) 255489.7226 661137.9182 0.386439373 0.699580064BV_P43_V_9..3 -140.5508408 314.737018 -0.446565967 0.65566922CONTI_M1_OHM.144 0.033316344 0.019782193 1.684158309 0.093703293CONTI_M2_OHM.18 0.039386224 0.044631996 0.882466111 0.378578943CONTI_M3_OHM.18 0.026814633 0.044482287 0.602815974 0.547310648CONTI_M4_OHM.18 -0.045582212 0.040562664 -1.123747997 0.262460274CONTI_M5_OHM.18 0.017780782 0.041975094 0.423603149 0.672308197CONTI_M6_OHM.18 0.577014724 0.564469955 1.022223981 0.307903632CONTI_PO_OHM.117 0.004428242 0.001916507 2.310579203 0.021870687Isat_N43_mA_9..35 -5.357995572 31.64198546 -0.169331839 0.865705927Isat_N4_mA_9..117 43.33439442 22.89066542 1.893103307 0.059780909Isat_P43_mA_9..3 27.85740464 45.68855502 0.609723915 0.542733355Isat_P4_mA_9..117 31.79755619 37.19205818 0.85495554 0.393593667MIM_CAP_fF.um2 74.6790648 71.73786455 1.040999272 0.299126412Rc_N.PO_OHM.SE.144 -11.29487994 24.49504305 -0.461108801 0.645219108Rc_N._OHM.SE.144 -8.631379545 13.60267605 -0.634535404 0.526453025Rc_P.PO_OHM.SE.144 15.65932735 22.43255259 0.698062661 0.48594478Rc_P._OHM.SE.144 -1.816391911 13.85401223 -0.131109449 0.895819859
Residual standard error: 6.297 on 201 degrees of freedomMultiple R-Squared: 0.4439, Adjusted R-squared: 0.3111 F-statistic: 3.343 on 48 and 201 DF, p-value: 1.455e-09
Improve the model?
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Testing Hypothesis
Divided into two group by yield Null: mean equal For all parameters
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Result:Parameter p-valueBV_N43_V_9..35 1BV_N4_V_9..117 1BV_P43_V_9..3 0.028716BV_P4_V_9..117 1CONTI_M1_OHM.144 0.021956CONTI_M2_OHM.18 0.001909CONTI_M3_OHM.18 0.016357CONTI_M4_OHM.18 0.128322CONTI_M5_OHM.18 0.299886CONTI_M6_OHM.18 4.15E-06CONTI_PO_OHM.117 0.309631Isat_N43_mA_9..35 0.038173Isat_N4_mA_9..117 0.026588Isat_P43_mA_9..3 7.29E-10Isat_P4_mA_9..117 0.858306MIM_CAP_fF.um2 0.19949Rc_N.PO_OHM.SE.144 0.046481Rc_N._OHM.SE.144 0.015235
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Display Graphics
More than two group?Other display way?
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Fab 廠資料來源
量測參數資料 (wafer) 製造時機台的監控資料 (equipment) Defect inspection WAT CP Wafer map( 由 Defect/WAT/CP 所衍生出來
的 )
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Fab 廠的挑戰
Time to market
-- yield
-- new technology
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Fab 廠常用的統計方法Statistical Method Purpose
Distribution Basic material for statistical tests. Used to characterize a population based upon a sample.
Hypothesis testing Decide whether data under investigation indicates that elementsof concern are the “same” or “different.”
Experimental design andanalysis of variance
Determine significance of factors and models;Decompose observed variation into constituent elements.
Categorical modelingUse when result or response is discrete (such as “very rough,”“rough,” or “smooth”). Understand relationships, determineprocess margin, and optimize process.
Statistical process control Determine if system is operating as expected.
Regression Yield modeling. Yield impact
Duane S. Boning, Jerry Stefani and Stephanie W. Butler: Statistical Methods for Semiconductor manufacturing
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Yield maintenance
Process stable
-- statistical process control Excursion resolve
-- finding root cause of yield drop PM
-- Preventative maintenance
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Statistical Process Control
Normal, +- 3 sigma ~ 99.7
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Yield drop
process equipment malfunction
-- which process, which equipment(s)
group comparison for all possible process
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Group comparison
Null: mean equal for all groups Alternative: mean not equal Group by equipment Mean of measurement data
Other methods?
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PM
Why?
損耗 污染 When
經驗值 原始設定值 better way?
first wafer effect
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Yield enhancement
DOE for process improve Finding key parameters for yield Yield impact model
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Process improvement
Material Processing time
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Key parameters
Domain knowledge
物理性質 Regression
variables
collinear PCA
Other method?
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Yield impact model
問題 -- Defect item/Pattern
對 yield 的影響有多大 資料
Lot Wafer Yield Pattern1 Pattern2 Pattern3 Pattern4 Pattern5LA001 WA01 88 0 1 0 0 0LA001 WA02 89 0 0 0 0 0LA001 WA03 85 0 1 1 0 0LA001 WA04 75 0 0 1 0 0LA001 WA05 84 0 1 0 1 0LA002 WA01 82 0 1 0 1 0LA002 WA02 88 0 1 0 1 0LA002 WA03 87 0 1 0 0 0LA002 WA04 86 0 0 0 1 0LA002 WA05 85 0 1 0 0 0LA003 WA01 88 0 0 0 1 0LA003 WA02 84 0 0 0 0 0LA003 WA03 85 0 0 1 1 0LA003 WA04 83 0 0 0 1 0LA003 WA05 87 0 0 1 0 0LA004 WA01 92 1 0 0 0 0LA004 WA02 85 0 0 1 0 0LA004 WA03 87 1 0 1 0 0
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Yield impact model
Logistic regression
coefficient as “kill probability”
impact value = pattern loss/total loss
每片 wafer 的 yield loss 依 kill probability 的比率分給每個有的 pattern, 總合所有的 wafer就是該 pattern 的 pattern loss
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Yield Prediction
Wafer Yield
Week/month
Next product
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Wafer Yield -- Poisson Model
D : chip defect density A : critical Area (chip area)
ADY e
Assumption: n defects randomly distribute in wafer with N chips
The probability of one chip contains k defects
!
km
k
mP e
k m=n/N,
K=0 Chip passD = m/A Yield = Pass chip number/N
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Wafer Yield -- Other models
Non-uniform defect density
0
( )ADY e f D dD
f(D) is the defect density distribution
Murphy
model density formulation
triangular2
1 ADeY
AD
Exponential 1
1Y
AD
Seeds ADY e21
2
ADeY
AD
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Daily/Weekly/Monthly Yield
Average of total wafers? Regression
-- parameter selections
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Next product Yield
Technology baseline Previous product yield
Time
Yield
Pilot Rump
Mass production Phase out
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Summary of “basic statistical method” Statistics: description, display Testing hypotheses:
root cause, important parameters Regression:
yield modeling
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新的挑戰 -- 進階分析的方法
資料量 變數維度 更好的控制 演算速度的提昇 跨領域的合作
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Advanced topic -- testing
Null hypothesis
當 n 夠大時容易 reject
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Display – violin plot
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Display -- Correlogram
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Pattern classify
Question
-- Which category is this wafer belong to? Example
--
Ref: pattern recognition
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Pattern Classify
如何描述 pattern
共有但可以區別的特性
距離 怎麼定
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Clustering 分群
那幾片 wafer 可當成同一類 怎麼分 --Hierarchical
up-down/bottom-up
--Partitional
k-means/k-mediods
Reference: http://en.wikipedia.org/wiki/Cluster_analysis#Hierarchical_clustering
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Single Tool
Testing? Trend
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Golden Path
P1
P2
P3
E1
E2
T1
T2
3*2*2 = 12 combination for 3 steps Whole combination?
Partition methodGolden wafer/golden lot tracking
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Parameters >> observations
Grouping parameters Supersaturated design analysis
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Advanced Process Control
Run-to-run
-- feedback control Fault detection
-- Abnormal Virtual Metrology
-- reduce metrology
-- feed for r2r
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Reference:
http://www.siliconfareast.com/test-yield-models.htm http://www.icyield.com/yieldmod.html