統計在半導體產業的應用 -- basic statistic methods

半導體產業常用的統計方法

大綱

常用的統計方法 各產業的挑戰 進階分析用的方法

基本統計量

Describe statistics

-- mean/median/std/Q1/Q3/percentile Display

-- box/histogram/CDF/P-P/Q-Q

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

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 ?

可用的統計方法

Correlation coefficient Regression Testing hypothesis

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

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

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?

Testing Hypothesis

Divided into two group by yield Null: mean equal For all parameters

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

Display Graphics

More than two group?Other display way?

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 )

Fab 廠的挑戰

Time to market

-- yield

-- new technology

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

Yield maintenance

Process stable

-- statistical process control Excursion resolve

-- finding root cause of yield drop PM

-- Preventative maintenance

Statistical Process Control

Normal, +- 3 sigma ~ 99.7

Yield drop

process equipment malfunction

-- which process, which equipment(s)

group comparison for all possible process

Group comparison

Null: mean equal for all groups Alternative: mean not equal Group by equipment Mean of measurement data

Other methods?

PM

Why?

損耗 污染 When

經驗值 原始設定值 better way?

first wafer effect

Yield enhancement

DOE for process improve Finding key parameters for yield Yield impact model

Process improvement

Material Processing time

Key parameters

Domain knowledge

物理性質 Regression

variables

collinear PCA

Other method?

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

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

Yield Prediction

Wafer Yield

Week/month

Next product

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

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

Daily/Weekly/Monthly Yield

Average of total wafers? Regression

-- parameter selections

Next product Yield

Technology baseline Previous product yield

Time

Yield

Pilot Rump

Mass production Phase out

Summary of “basic statistical method” Statistics: description, display Testing hypotheses:

root cause, important parameters Regression:

yield modeling

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

Advanced topic -- testing

Null hypothesis

當 n 夠大時容易 reject

Display – violin plot

Display -- Correlogram

Pattern classify

Question

-- Which category is this wafer belong to? Example

--

Ref: pattern recognition

Pattern Classify

如何描述 pattern

共有但可以區別的特性

距離 怎麼定

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

Single Tool

Testing? Trend

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

Parameters >> observations

Grouping parameters Supersaturated design analysis

Advanced Process Control

Run-to-run

-- feedback control Fault detection

-- Abnormal Virtual Metrology

-- reduce metrology

-- feed for r2r

Reference:

http://www.siliconfareast.com/test-yield-models.htm http://www.icyield.com/yieldmod.html